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AVERAGES OF TWISTED $L$-FUNCTIONS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  24 June 2015

JULIA JACKSON  and
ANDREW KNIGHTLY
JULIA JACKSON
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, USA email j.jackson@ou.edu
ANDREW KNIGHTLY*
Affiliation:
Department of Mathematics and Statistics, University of Maine, Orono, ME 04469-5752, USA email knightly@math.umaine.edu
*
✉knightly@math.umaine.edu
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Abstract

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We use a relative trace formula on $\text{GL}(2)$ to compute a sum of twisted modular $L$-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight $k$ or level $N$ is sufficiently large, the sum is nonzero. Specializing to the central point, we show in some cases that the resulting bound for the average is as good as that predicted by the Lindelöf hypothesis in the $k$ and $N$ aspects.

Keywords

relative trace formulaL-functionsmodular forms

MSC classification

Primary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 99 , Issue 2 , October 2015 , pp. 207 - 236
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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