  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.
