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.

中文翻译：

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L 函数族的加权单级密度

本文致力于非平凡零点的单级密度的加权版本$L$- 函数，由 的力量倾斜$L$-在中心点评估功能。假设黎曼假设和比率猜想，对于某些特定族$L$-函数，我们证明，如果权重的指数足够小，密度猜想所暗示的相同结构也适用于这个加权研究。此外，我们推测一般情况，推测加权核的显式公式。