The first moment of quadratic twists of modular $L$-functions
Volume 206 / 2022
Acta Arithmetica 206 (2022), 313-337
MSC: Primary 11M06; Secondary 11F67.
DOI: 10.4064/aa211207-7-11
Published online: 4 January 2023
Abstract
We obtain an asymptotic formula with an error term $O(X^{1/2+\varepsilon})$ for the smoothed first moment of quadratic twists of modular $L$-functions. We also give a similar result for the smoothed first moment of the first derivative of quadratic twists of modular $L$-functions. The argument is largely based on Young’s works [Acta Arith. 138 (2009), 73–99, and Selecta Math. 19 (2013), 509–543].
