Averaging quadratically twisted $L$-values and their derivatives

TL;DR

This paper derives an asymptotic formula for the product of quadratically twisted central $L$-values and derivatives associated with pairs of holomorphic cusp forms, extending previous work in the field.

Contribution

It unconditionally establishes a new asymptotic formula for products of twisted $L$-values and derivatives, expanding the understanding of their behavior.

Findings

Established an asymptotic formula for twisted $L$-value products

Extended previous results to include derivatives of $L$-values

Unconditional proof of the asymptotic behavior

Abstract

In this paper, we unconditionally establish an asymptotic formula for the product of the quadratically twisted central $L$-value associated to a holomorphic cusp form $f$, and the quadratically twisted central $L$-derivative to a distinct holomorphic cusp form $g$. This result may be viewed as an extension of \cite{Li-MR4768632}, \cite{Kumar.etc-MR4765788} and \cite{zhou2025momentderivativesquadratictwists}.