Algebra & Number Theory
- Algebra Number Theory
- Volume 12, Number 1 (2018), 35-59.
The mean value of symmetric square $L$-functions
We study the first moment of symmetric-square -functions at the critical point in the weight aspect. Asymptotics with the best known error term were obtained independently by Fomenko in 2003 and by Sun in 2013. We prove that there is an extra main term of size in the asymptotic formula and show that the remainder term decays exponentially in . The twisted first moment was evaluated asymptotically by Ng with the error bounded by . We improve the error bound to unconditionally and to under the Lindelöf hypothesis for quadratic Dirichlet -functions.
Algebra Number Theory, Volume 12, Number 1 (2018), 35-59.
Received: 20 October 2016
Revised: 30 June 2017
Accepted: 15 November 2017
First available in Project Euclid: 4 April 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11F12: Automorphic forms, one variable
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$ 34E05: Asymptotic expansions 34E20: Singular perturbations, turning point theory, WKB methods
Balkanova, Olga; Frolenkov, Dmitry. The mean value of symmetric square $L$-functions. Algebra Number Theory 12 (2018), no. 1, 35--59. doi:10.2140/ant.2018.12.35. https://projecteuclid.org/euclid.ant/1522807230