Estimates for Sums of Coefficients of Dirichlet Series with Functional Equation

Suppose we have a Dirichlet series L(s) = ∑n = 1∞ann−s such that it, and its twists by Dirichlet characters have analytic continuation and a functional equation of a specific kind. Suppose also that the root numbers of the twists are equidistributed on the unit circle. The purpose of this note is to get an estimate for the quantity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum\limits_{\mathop {n \leqslant x}\limits_{n \equiv \ell (\bmod p)} } {a_n - \frac{1}{{\phi (p)}}} \sum\limits_{\mathop {n \leqslant x}\limits_{(n,p){\text{ = }}1} } {a_n } $$ \end{document}for a prime modulus p.