Classification of L-functions of degree 2 and conductor 1

TL;DR

This paper fully classifies degree 2, conductor 1 L-functions within the extended Selberg class using a new invariant, confirming they are automorphic L-functions and extending classical converse theorems.

Contribution

Introduces a new numerical invariant $$ to characterize degree 2, conductor 1 L-functions, providing a sharp classification and confirming their automorphic nature.

Findings

The invariant $$ precisely describes the nature of these L-functions.

The classification confirms the automorphic conjecture for this class.

Provides a new framework for understanding L-functions via functional equation data.

Abstract

We give a full description of the functions $F$ of degree 2 and conductor 1 in the general framework of the extended Selberg class. This is performed by means of a new numerical invariant $\chi_F$, which is easily computed from the data of the functional equation. We show that the value of $\chi_F$ gives a precise description of the nature of $F$, thus providing a sharp form of the classical converse theorems of Hecke and Maass. In particular, our result confirms, in the special case under consideration, the conjecture that the functions in the Selberg class are automorphic $L$-functions.