Moments of central 𝐿-values for Maass forms over imaginary quadratic fields

Liu, Sheng-Chi; Qi, Zhi

doi:10.1090/tran/8588

Moments of central $L$-values for Maass forms over imaginary quadratic fields
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Abstract:

In this paper, over imaginary quadratic fields, we consider the family of $L$-functions $L (s, f)$ for an orthonormal basis of spherical Hecke–Maass forms $f$ with Archimedean parameter $t_f$. We establish asymptotic formulae for the twisted first and second moments of the central values $L\big (\frac 1 2, f\big )$, which can be applied to prove that at least $33 %$ of $L\big (\frac 1 2, f\big )$ with $t_f \leqslant T$ are non-vanishing as $T \rightarrow \infty$. Our main tools are the spherical Kuznetsov trace formula and the Voronoï summation formula over imaginary quadratic fields.

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