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Selberg's conjecture A states that whenever $F$ is an element of the Selberg class, there exists a non negative integer $n_F$ depending only on $F$ such that $\sum_{p\leq x}\dfrac{\vert a_{p}(F)\vert^{2}}{p}=n_F\log\log x+O(1)$, where $a_n(F)$ is the coefficient of the Dirichlet series defining $F(s)$ provided $\Re(s)>1$. If one only supposes that $a_n(F)=O_{\varepsilon}(n^{\varepsilon})$ and that there exists a non negative integer $m$ such that $(s-1)^m.F(s)$ is an entire function of finite order, what are the best results about the growth of the function $G_F:=x\mapsto\sum_{p\leq x}\dfrac{\vert a_{p}(F)\vert^{2}}{p}$? Can it be shown, for example, that this implies that $G_F(x)=O(\log^\alpha x)$ for some positive real number $\alpha$?
Thank you in advance.

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