A weighted one-level density of families of $L$-functions

Abstract

This paper is devoted to a weighted version of the one-level density of the non-trivial zeros of $L$-functions, tilted by a power of the $L$-function evaluated at the central point. Assuming the Riemann Hypothesis and the ratio conjecture, for some specific families of $L$-functions we prove that the same structure suggested by the density conjecture holds also in this weighted investigation, if the exponent of the weight is small enough. Moreover we speculate about the general case, conjecturing explicit formulae for the weighted kernels.

arXiv comment50 pages

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• Moments and distribution of central L-values of quadratic twists of elliptic curves
• The second moment of quadratic twists of modular $L$-functions
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Cited by in corpus (1)

• Weighted low-lying zeros of L-functions attached to Siegel modular forms