Zeros of L-functions outside the critical strip

Booker, Andrew; Thorne, Frank

doi:10.2140/ant.2014.8.2027

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Zeros of $L$-functions outside the critical strip

Andrew R. Booker and Frank Thorne

Vol. 8 (2014), No. 9, 2027–2042

DOI: 10.2140/ant.2014.8.2027

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 $f \in S_{k} (Γ_{1} (N))$ is a classical holomorphic modular form whose $L$ -function does not vanish for $ℜ (s) > (k + 1) ∕ 2$ , 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.

A correction was submitted 16 Mar 2018 and posted 6 Sep 2018 in an online supplement.

Keywords

$L$-functions, Euler products, automorphic forms

Mathematical Subject Classification 2010

Primary: 11F66

Secondary: 11M99, 11F11

Milestones

Received: 26 June 2013

Revised: 17 June 2014

Accepted: 25 August 2014

Published: 28 December 2014

Authors

Andrew R. Booker
School of Mathematics University of Bristol University Walk Bristol BS8 1TW United Kingdom

Frank Thorne
Department of Mathematics University of South Carolina 1523 Greene Street Columbia, SC 29208 United States

Vol. 8, No. 9, 2014