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arXiv · 2201.03856

Weighted average values of automorphic $L$-functions

Wei Liu
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Abstract

Let $S_2^*(q)$ be the set of primitive Hecke eigenforms of weight 2 and prime level $q$. For $p$ prime and $t\in \mathbb{R}$, we prove asymptotic formulas for the sums $$ \mathcal {A}(p^j,q,t)=\sum_{f\in S_2^*(q)} L\left(\frac{1}{2}+it,f\right)^2\lambda_f(p^j),\qquad j=1,2, $$ where $\lambda_f(p^j)$ is the $p^j$-th normalized Fourier coefficient of $f$ and $L(s,f)$ is the $L$-function associated to $f$.

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Wei Liu. 2022-01-11. Weighted average values of automorphic $L$-functions. https://arxiv.org/abs/2201.03856

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