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

TL;DR

This paper investigates a weighted one-level density of zeros of $L$-functions, assuming RH and the ratio conjecture, and extends the density conjecture to weighted cases with specific families.

Contribution

It introduces a weighted version of the one-level density for $L$-functions and proves the conjecture holds for certain families under specific conditions.

Findings

Weighted density matches the unweighted structure for small exponents

Results are conditional on RH and the ratio conjecture

Conjectures for explicit formulas in the general case

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.