We compute the first moment of twisted Hecke $L$-functions of weight $2$ and prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.
"The first moment of twisted Hecke $L$-functions with unbounded shifts." Funct. Approx. Comment. Math. 57 (1) 105 - 114, September 2017. https://doi.org/10.7169/facm/1624