Weil's converse theorem for Maass forms and cancellation of zeros

TL;DR

This paper characterizes Maass forms via functional equations with meromorphically twisted Dirichlet series and demonstrates that a specific quotient of L-functions has infinitely many poles, advancing understanding of automorphic forms.

Contribution

It introduces a new characterization of Maass forms using meromorphic twists and proves the existence of infinitely many poles in a related L-function quotient.

Findings

Characterization of Maass forms through functional equations with meromorphic twists

Proof that the quotient of the symmetric square L-function and zeta function has infinitely many poles

Extension of Weil's converse theorem to Maass forms with meromorphic twists

Abstract

We prove two principal results. Firstly, we characterise Maass forms in terms of functional equations for Dirichlet series twisted by primitive characters. The key point is that the twists are allowed to be meromorphic. This weakened analytic assumption applies in the context of our second theorem, which shows that the quotient of the symmetric square L-function of a Maass newform and the Riemann zeta function has infinitely many poles.