Algebra & Number Theory

Zeros of $L$-functions outside the critical strip

Andrew Booker and Frank Thorne

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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 fSk(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.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Secondary: 11M99: None of the above, but in this section 11F11: Holomorphic modular forms of integral weight

$L$-functions Euler products automorphic forms


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.

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