## Algebra & Number Theory

### Zeros of $L$-functions outside the critical strip

#### 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 $ℜ(s)>(k+1)∕2$, 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.

#### Article information

Source
Algebra Number Theory, Volume 8, Number 9 (2014), 2027-2042.

Dates
Received: 26 June 2013
Revised: 17 June 2014
Accepted: 25 August 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730308

Digital Object Identifier
doi:10.2140/ant.2014.8.2027

Mathematical Reviews number (MathSciNet)
MR3294385

Zentralblatt MATH identifier
1320.11044

#### Citation

Booker, Andrew; Thorne, Frank. Zeros of $L$-functions outside the critical strip. Algebra Number Theory 8 (2014), no. 9, 2027--2042. doi:10.2140/ant.2014.8.2027. https://projecteuclid.org/euclid.ant/1513730308

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