General L-functions and the Selberg Class

By now you have met a small zoo of "zeta-like" objects: the Riemann zeta function \zeta(s)=\sum_{n\ge1} n^{-s}, the Dirichlet L-functions L(s,\chi)=\sum_{n\ge1}\chi(n)\,n^{-s}, and — lurking on the horizon — the L-functions attached to modular forms and to elliptic curves. Stare at them long enough and something uncanny happens: they all wear the same uniform.

Each one is a Dirichlet series that converges in a half-plane; each has an Euler product over the primes; each continues to a meromorphic function on the whole plane; and each satisfies a functional equation relating s to 1-s through a bundle of Gamma factors. In 1989–92, Atle Selberg asked the natural question: what if we stop treating these as separate miracles and instead write down the axioms they share — then study every function that obeys them? The result is the Selberg class \mathcal S, and it turns the theory of L-functions from a bestiary into a single, structured universe.

The shape they all share

Before the axioms, let's name the common silhouette. A "general L-function" is built from three ingredients stacked together:

F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^{s}}\quad(\text{a Dirichlet series}),\qquad F(s)=\prod_{p}\big(\text{local factor at }p\big)\quad(\text{an Euler product}),

together with a completed function \Phi(s) — the series multiplied by a Gamma factor and a power of a conductor — that satisfies a clean reflection \Phi(s)=\omega\,\overline{\Phi}(1-s). The Euler product encodes the primes; the Dirichlet series makes it an analytic object; the functional equation gives it a mirror symmetry. Selberg's insight was that these three, pinned down carefully, are enough to force a huge amount of structure — including, conjecturally, a Riemann Hypothesis for every member.

The five axioms

Here is the definition in full. A function F(s) of the complex variable s=\sigma+it belongs to the Selberg class \mathcal S exactly when it satisfies all five of the following.

(1) Dirichlet series. F(s)=\sum_{n=1}^{\infty} a_n\,n^{-s} with a_1=1, absolutely convergent for \Re(s)>1.
(2) Analytic continuation. There is an integer m\ge 0 such that (s-1)^{m}F(s) extends to an entire function of finite order (so the only possible pole is at s=1).
(3) Functional equation. The completed function \Phi(s)=Q^{s}\prod_{i=1}^{r}\Gamma(\lambda_i s+\mu_i)\,F(s) satisfies \Phi(s)=\omega\,\overline{\Phi}(1-s), where Q>0, \lambda_i>0, \Re(\mu_i)\ge 0, |\omega|=1, and \overline{\Phi}(s)=\overline{\Phi(\bar s)}.
(4) Ramanujan bound. The coefficients grow slowly: a_n\ll_\varepsilon n^{\varepsilon} for every \varepsilon>0.
(5) Euler product. \log F(s)=\sum_{n=1}^{\infty} b_n\,n^{-s}, where b_n=0 unless n=p^{k} is a prime power, and b_n\ll n^{\theta} for some \theta<\tfrac12.

Read them slowly. Axiom (1) says "it's a Dirichlet series, normalised so the first coefficient is 1". Axiom (2) says "it's meromorphic with at worst one pole, at s=1". Axiom (3) is the reflection through s\mapsto 1-s that you already know from \zeta. Axiom (4) forbids the coefficients from blowing up. Axiom (5) says \log F is supported on prime powers — which is exactly what an Euler product \prod_p(\dots) gives when you take its logarithm.

The degree — a single number that classifies everything

The most important invariant reads straight off the functional equation. The Gamma factors in axiom (3) come with numbers \lambda_i; their doubled sum is the degree:

The degree is d_F=2\sum_{i=1}^{r}\lambda_i.
Although the factorisation of the Gamma factors is not unique, the quantity d_F is a well-defined invariant of F.
Every known member has degree a non-negative integer, and this is conjectured to hold always.

The degree measures "how many Gamma factors" the completed function needs — loosely, the dimension of the arithmetic object standing behind F. It is the organising parameter of the whole class: you classify Selberg-class functions first by degree, then within each degree. And here is the punchline that makes the theory sharp — the degree is quantised.

Degree 0: the only element is the constant function F\equiv 1.
No degrees strictly between 0 and 1: there is nothing with 0 (Conrey–Ghosh, Kaczorowski–Perelli).
Degree 1: exactly the Riemann zeta function and the (shifted) Dirichlet L-functions of primitive characters — nothing else.

So the degrees observed are 0,1,2,3,\dots, with visible gaps. It is a striking fact that a purely analytic set of axioms should force such an arithmetic, integer-valued ladder — as if the class could only be built out of whole "bricks".

Worked example — verifying that \zeta lives in \mathcal S

Let's check the flagship member against the five axioms and read off its degree.

(1) Dirichlet series. \zeta(s)=\sum_{n\ge1} n^{-s} has a_n=1 for all n, so a_1=1 and the series converges absolutely for \Re(s)>1 (compare with \sum n^{-\sigma}). ✓

(2) Continuation. \zeta continues to a meromorphic function whose only pole is a simple pole at s=1, so (s-1)\zeta(s) is entire of finite order — take m=1. ✓

(3) Functional equation. The completed zeta function

\Phi(s)=\pi^{-s/2}\,\Gamma\!\left(\tfrac{s}{2}\right)\zeta(s) \quad\text{satisfies}\quad \Phi(s)=\Phi(1-s).

This matches axiom (3) with a single Gamma factor: Q=\pi^{-1/2}, r=1, \lambda_1=\tfrac12, \mu_1=0, and root number \omega=1 (and \overline{\Phi}=\Phi since the coefficients are real). ✓

(4) Ramanujan bound. Here a_n=1\ll n^{\varepsilon} trivially. ✓

(5) Euler product. From \zeta(s)=\prod_p (1-p^{-s})^{-1} we get \log\zeta(s)=-\sum_p\log(1-p^{-s})=\sum_p\sum_{k\ge1}\tfrac1k\,p^{-ks}, so b_n=\tfrac1k when n=p^{k} and b_n=0 otherwise — supported on prime powers, and bounded (take any \theta<\tfrac12). ✓

All five hold, so \zeta\in\mathcal S. Its degree is

d_\zeta=2\sum_i\lambda_i=2\cdot\tfrac12=1.

The single \Gamma(s/2) — one Gamma factor with \lambda=\tfrac12 — is precisely what makes \zeta a degree-1 L-function.

A gallery of examples, by degree

Each classical L-function slots into the class at a definite degree, set by how many Gamma factors its functional equation needs. Here are the headline cases.

L-function	Coefficients a_n	Gamma factors	Degree d
Riemann zeta \zeta(s)	a_n=1	\Gamma(s/2)	1
Dirichlet L(s,\chi), \chi primitive	a_n=\chi(n)	\Gamma\!\big(\tfrac{s+\mathfrak a}{2}\big)	1
Modular form L(s,f) (holomorphic newform)	Hecke eigenvalues a_n	\Gamma\!\big(s+\tfrac{k-1}{2}\big)	2
Elliptic curve L(s,E)	a_p=p+1-\#E(\mathbb F_p)	\Gamma(s) (analytic normalisation)	2
Dedekind zeta \zeta_K(s), [K:\mathbb Q]=n	ideal-counting	n Gamma factors	n

Notice the pattern: degree 1 is the world of \zeta and Dirichlet characters (one-dimensional objects); degree 2 is the world of modular forms and elliptic curves (which, by modularity, are secretly the same degree-2 L-functions); and a number field of degree n gives a degree-n zeta function. The degree is a bridge between the analytic object and the arithmetic it came from.

An elliptic curve E feeds you one number per prime, a_p=p+1-\#E(\mathbb F_p) — so you might guess its L-function looks one-dimensional, like a Dirichlet L. But the local factor at a good prime is a quadratic in p^{-s}, (1-a_p p^{-s}+p^{1-2s})^{-1}, with two reciprocal roots \alpha_p,\beta_p (the Hasse bound gives |\alpha_p|=|\beta_p|=\sqrt p). Two roots per prime is the signature of degree 2, and it forces two Gamma factors' worth of archimedean data (one \Gamma(s) in the analytic normalisation, i.e. 2\lambda=2\cdot 1). Modularity then says this degree-2 L-function is equal to the L-function of a weight-2 modular form — the same object seen from two sides.

Primitivity, factorisation, and orthonormality

Just as every whole number factors uniquely into primes, the Selberg class has its own "prime" elements. Call F\in\mathcal S, F\ne 1, primitive if it cannot be written as a product F=F_1F_2 of two non-trivial members of \mathcal S.

Every F\in\mathcal S factors as a product of primitive functions.
The degree is additive under multiplication: d_{FG}=d_F+d_G.
Because a degree-1 element cannot split (you cannot write 1=d_{F_1}+d_{F_2} with both parts \ge 1), every degree-1 element — in particular \zeta and each primitive Dirichlet L — is primitive.

Whether this factorisation is unique is one of the central open questions, and it is tied to a beautiful conjecture about how the coefficients of two primitive functions correlate.

For every primitive F, \displaystyle\sum_{p\le x}\frac{|a_p(F)|^{2}}{p}=\log\log x+O(1).
For two distinct primitive F\ne G, \displaystyle\sum_{p\le x}\frac{a_p(F)\,\overline{a_p(G)}}{p}=O(1).

In words: the prime-coefficients of distinct primitive L-functions are "orthogonal", and each is "normalised" — hence orthonormality. Selberg conjectured this, and it is known to imply unique factorisation of \mathcal S into primitives, making the class behave like a genuine analogue of the integers. It is expected to match the notion of an automorphic L-function, where distinctness corresponds to inequivalent cuspidal representations.

The Grand Riemann Hypothesis

The single most consequential conjecture about the class is the wholesale generalisation of what you met for \zeta.

Every F\in\mathcal S has all of its non-trivial zeros on the critical line \Re(s)=\tfrac12.

This one statement swallows the classical Riemann Hypothesis (take F=\zeta), the Generalised Riemann Hypothesis for Dirichlet L-functions (take F=L(s,\chi)), and the corresponding hypotheses for modular and elliptic-curve L-functions, all at once. The Selberg class is, in effect, the largest natural family for which one can even state a Riemann Hypothesis — which is a large part of why Selberg wrote the axioms down. Every consequence you proved "assuming RH" or "assuming GRH" becomes a consequence of GRH across the entire class.

A tempting misreading is "the Selberg class is just the set of Dirichlet series with a functional equation". It is not. Membership demands all five axioms simultaneously — and the last two do real work. Drop the Ramanujan bound (4) or the Euler product (5) and you let in pathological series that share the analytic shape of an L-function but none of its arithmetic. It is the Euler product, in particular, that ties F to the primes and makes degree and primitivity meaningful. A functional equation alone is not enough.

A second, subtler caution: it is conjectured, but not proved, that the Selberg class coincides with the automorphic L-functions (those coming from cuspidal representations of \mathrm{GL}_n). Every automorphic L-function is expected to lie in \mathcal S, and every element of \mathcal S is expected to be automorphic — but proving the two collections are literally the same set is wide open. The Selberg class is an axiomatic frame around the automorphic world, not a theorem that they agree.

The quantisation results are not soft. Conrey and Ghosh (1993) proved there is no element with 0; Kaczorowski and Perelli later pushed the classification to show that the only degree-1 members are \zeta and shifts of primitive Dirichlet L-functions, and that there is nothing at all with 1. The arguments run by feeding the functional equation and the Euler product into each other: the Euler product forces the coefficients to be multiplicative-ish and non-negative in log, while the functional equation with small total \sum\lambda_i is too rigid to support them unless the whole thing degenerates to something classical. The moral: axioms (3) and (5) together are far more constraining than either alone — which is exactly why the degree can only land on the integer rungs.