Algebra & Number Theory
- Algebra Number Theory
- Volume 8, Number 9 (2014), 2027-2042.
Zeros of $L$-functions outside the critical strip
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 is a classical holomorphic modular form whose -function does not vanish for , then is a Hecke eigenform. Our proof adapts and extends work of Saias and Weingartner, who proved a similar result for degree- -functions.
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
Permanent link to this document
Digital Object Identifier
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
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