Averaging quadratically twisted modular L -values and their derivatives

Abstract In this paper, we unconditionally establish an asymptotic formula for the product of the central value of the quadratically twisted L -function associated with a holomorphic cusp form f , and the central value of the derivative of the quadratically twisted L -function with a distinct holomorphic cusp form g . This result may be viewed as an extension of [X. Li, Moments of quadratic twists of modular L -functions, Invent. Math. 237 2024, 2, 697–733], [S. Kumar, K. Mallesham, P. Sharma and S. K. Singh, Moments of derivatives of modular L -functions, Q. J. Math. 75 2024, 2, 715–734] and [Z. Zhou, Moment of derivatives of quadratic twists of modular l -functions, preprint 2025, https://arxiv.org/abs/2503.14680 ].