The first moment of quadratic twists of modular $L$-functions

Abstract

We obtain an asymptotic formula with an error term $O(X^{\frac{1}{2} + \varepsilon})$ 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(1):73-99, 2009 and Selecta Math 19(2):509-543, 2013).