We prove that the central values of additive twists of a cuspidal L-function define a quantum modular form in the sense of Zagier, generalizing recent results of Bettin and Drappeau. From this, we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal L-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet L-functions. Furthermore, we give an interpretation of quantum modularity at infinity for additive twists of L-functions of weight 2 cusp forms in terms of the corresponding functional equations.

中文翻译：

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关于加性扭曲，互易定律和量子模态形式的注释

我们证明了尖齿L函数的加性扭曲的中心值在Zagier的意义上定义了量子模块形式，概括了Bettin和Drappeau的最新结果。由此，我们推导出了针对尖齿L函数乘性扭曲的扭曲第一矩的互易定律，类似于Conrey针对Dirichlet L函数扭曲第二矩的互易定律。此外，根据相应的函数方程，我们给出了权重2尖点形式的L函数的加性扭曲在无穷远处的量子模块化的解释。