Weighted averages of modular 𝐿-values

Knightly, Andrew; Li, Charles

doi:10.1090/S0002-9947-09-04923-X

Weighted averages of modular $L$-values
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by Andrew Knightly and Charles Li

Trans. Amer. Math. Soc. 362 (2010), 1423-1443

DOI: https://doi.org/10.1090/S0002-9947-09-04923-X

Published electronically: September 25, 2009

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Abstract:

Using an explicit relative trace formula on $\operatorname {GL}(2)$, we derive a formula for averages of modular $L$-values in the critical strip, weighting by Fourier coefficients, Hecke eigenvalues, and Petersson norms. As an application we show that a GRH holds for these averages as the weight or the level goes to $\infty$. We also use the formula to give explicit zero-free regions of the form $|\operatorname {Im}(s)|\le \tau _0$ for some particular modular $L$-functions.

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