#### Vol. 8, No. 9, 2014

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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
##### 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 $f\in {S}_{k}\left({\Gamma }_{1}\left(N\right)\right)$ is a classical holomorphic modular form whose $L$-function does not vanish for $\Re \left(s\right)>\left(k+1\right)∕2$, 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
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