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

This paper is devoted to a weighted version of the one-level density of the nontrivial 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 also holds 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.