Vol. 8, No. 9, 2014

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Zeros of $L$-functions outside the critical strip

Andrew R. Booker and Frank Thorne

Vol. 8 (2014), No. 9, 2027–2042
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 Sk(Γ1(N)) is a classical holomorphic modular form whose L-function does not vanish for (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.

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