A converse theorem for degree 2 elements of the Selberg class with restricted gamma factor

TL;DR

This paper establishes a converse theorem characterizing degree 2 L functions with specific gamma factors, showing they correspond to either newforms or products of Dirichlet character L functions, under certain analytic conditions.

Contribution

It provides a new converse theorem for degree 2 L functions with restricted gamma factors, linking them to known automorphic forms without shape restrictions.

Findings

L functions coincide with those from newforms or Dirichlet characters

Allows arbitrary poles in twisted L functions

Requires analytic data on Euler factors and functional equations

Abstract

We prove a converse theorem for a family of L functions of degree 2 with gamma factor coming from a holomorphic cuspform. We show these L functions coincide with either those coming from a newform or a product of L functions arising from Dirichlet characters. We require some analytic data on the Euler factors, but don't require anything on the shape. We also suppose that the twisted L functions satisfy expected functional equations. We allow the non-trivial twists to have arbitrary poles.