Overview
This paper presents an asymptotic formula for the moment of the first-order derivative of modular $L$-functions. The evaluation occurs at the center of the critical strip. The derivatives are weighted by generalized divisor functions that are constructed using primitive quadratic characters. This work contributes to an area of study where such asymptotic formulae have been primarily established for higher-order derivatives or under the specific condition of dihedral forms. The moments under investigation align with prior research by Munshi and are relevant to the study of elliptic fibration.
Research Context
The study focuses on moments of derivatives of modular $L$-functions, specifically their first-order derivatives. These L-functions are evaluated at the central point of the critical strip. A distinguishing characteristic of this particular investigation is the weighting applied to these derivatives: generalized divisor functions, which are explicitly formed by primitive quadratic characters. Previous work by Munshi has identified that such moments naturally emerge within the context of elliptic fibration studies. Prior to this research, asymptotic formulae for these specific types of moments, particularly with the described weighting and for first-order derivatives, were not available. Existing asymptotic results in the literature were typically confined to cases involving higher-order derivatives or to situations where the forms were specialized to be dihedral.
Findings
The research successfully establishes an asymptotic formula for the moment of the first-order derivative of modular $L$-functions. This formula applies when these derivatives are assessed at the center of the critical strip. The weighting mechanism employed for these derivatives is based on generalized divisor functions, which are explicitly derived from primitive quadratic characters. The derivation of this asymptotic formula addresses a previously identified gap in the existing body of knowledge regarding moments of $L$-functions. Prior to this work, similar asymptotic formulae had been confined to two specific scenarios: either for higher-order derivatives of $L$-functions or for cases where the underlying forms were specialized to be dihedral forms. This finding therefore extends the applicability of asymptotic formulae to the first-order derivative in a more general nonlinear family setting, characterized by the specified weighting functions.