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September 2016 Nonvanishing and central critical values of twisted $L$-functions of cusp forms on average
Markus Schwagenscheidt
Funct. Approx. Comment. Math. 55(1): 45-58 (September 2016). DOI: 10.7169/facm/2016.55.1.4

Abstract

Let $f$ be a cusp form of integral weight $k \geq 4$ for $\Gamma_{0}(N)$ with nebentypus $\psi$. Generalising work of Kohnen we construct a kernel function for the $L$-function $L(f,\chi,s)$ of $f$ twisted by a primitive Dirichlet character $\chi$ and use it to show that the average $\sum_{f \in S_{k}(N,\psi)}\frac{L(f,\chi,s)}{\langle f,f\rangle}\overline{a_{f}(1)}$ over an orthogonal basis of $S_{k}(N,\psi)$ does not vanish on certain rectangles inside the critical strip if the weight $k$ or the level $N$ is big enough. As another application of the kernel function we prove an averaged version of Waldspurger's Theorem.

Citation

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Markus Schwagenscheidt. "Nonvanishing and central critical values of twisted $L$-functions of cusp forms on average." Funct. Approx. Comment. Math. 55 (1) 45 - 58, September 2016. https://doi.org/10.7169/facm/2016.55.1.4

Information

Published: September 2016
First available in Project Euclid: 19 September 2016

zbMATH: 06862552
MathSciNet: MR3549012
Digital Object Identifier: 10.7169/facm/2016.55.1.4

Subjects:
Primary: 11F03
Secondary: 11F67

Rights: Copyright © 2016 Adam Mickiewicz University

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Vol.55 • No. 1 • September 2016
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