Zeros of L-functions outside the critical strip

TL;DR

This paper investigates zeros of L-functions outside the critical strip, demonstrating that certain non-Euler product L-functions must have zeros in their region of absolute convergence, with implications for modular forms and eigenforms.

Contribution

It extends previous work by Saias and Weingartner to a broader class of automorphic L-functions, establishing new zero distribution results.

Findings

L-functions without Euler products have zeros in the region of absolute convergence.

If a modular form's L-function does not vanish beyond a certain line, it must be a Hecke eigenform.

The methods adapt and extend existing techniques to higher-degree L-functions.

Abstract

For a wide class of Dirichlet series associated to automorphic forms, we show that those without Euler products must have zeros within the region of absolute convergence. For instance, we prove that if f is a classical holomorphic modular form whose L-function does not vanish for Re(s) > (k+1)/2, then f is a Hecke eigenform. Our proof adapts and extends work of Saias and Weingartner, who proved a similar result for degree 1 L-functions.