We are concerned with an estimate and a mean square theorem for the summatory function of a class of Dirichlet series. This extension of Selberg's class is a class of Dirichlet series satisfying a functional equation involving multiple gamma factors and, contrary to the class studied by Chandrasekharan and Narasimhan, a conjugate, which allows twisted functions to belong to this class. If $F(s)=\sum_{n=1}^{+\infty}a_{n}n^{-s}$ is a Dirichlet series satisfying such a functional equation and $E(x)$ is the associated error term, then we prove $O$-estimate for $E(x)$ and $\int_{0}^{x}|E(y)|^2dy$. These results are similar to those of Chandrasekharan and Narasimhan but are applicable in cases where theirs are not.
