L-functions/SelbergClass - L-Functions Wiki

The Selberg class is an attempt at an axiomatic definition of what constitutes a L-function, regardless of its geometric origin.

Contents

Selberg Class Definition
Selberg Data
List by degrees

Selberg Class Definition

Defining the Selberg Class

This page gives the definition of the Selberg class, properties provable for elements, Selberg's and others conjectures, and consequences of these conjectures. References to be added.

Selberg Data

Conjecturally, the functional equation for an L-function will always take the form

\Phi(s)=\epsilon\cdot\overline{\Phi(1-\bar{s})},

where

\Phi(s):= q^{s/2} \prod_{j=1}^d \Gamma_{\mathbb{R}}(s+\mu_j) L(s).

\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)

This means each L-function can be (non-uniquely) associated to its Selberg Data, that is the 4-tuple

\bigg(d,q,\bigg\{\mu_i\bigg\},\epsilon\bigg).

Here

d is the degree of the L-function.,
q is the level of the L-function and should be an integer.
\{\mu_j\} are the Langlands parameters of the L-function.
\epsilon is the root number and satisfies |\epsilon|=1.

Selberg Class Definition

Selberg Data

List by degrees