On some analytic properties of a function associated with the Selberg class satisfying certain special conditions (Q6621596)

The extended Selberg class \({\mathcal S}\) is the class of non identically vanishing Dirichlet series \N\[\NF(s)=\sum_{n=1}^{+\infty}\frac{a_{F}(n)}{n^{s}}, \qquad \Re(s)>1 \N\]\Nabsolutely convergent for \(\Re(s)>1\) and such that the following conditions are satisfied: (a) \((s-1)^{m}F(s)\) is an entire function for some \(m\geq0\); (b) \(F(s)\) satisfies the functional equation \(\phi_{F}(s)=\omega\overline{\phi_{F}(1-\overline{s})},\) where \(|\omega|=1,\ \overline{f}(s)=\overline{f(\overline{s})}\) and \N\[\N\phi_{F}(s)=F(s)Q_{F}^{s}\prod_{j=1}^{r}\Gamma(\lambda_{j}s+\mu_{j})\N\]\Nwith \(r\geq0,\ Q>0,\ \lambda_{j}>0\) and \(\Re(\mu_{j}\geq0\); (c) for all \(\epsilon>0\); and (d) \(F(s)=\prod_{p}\exp\left(\sum_{l=0}^{\infty}\frac{b_{F}(p^{l})}{p^{ls}}\right)\) for \(\Re(s)>1\), where \(b_{F}(n)=0\) unless \(n=p^{m}\) with \(m\geq1\) and \(b_{F}(n)\ll n^{\theta}\) for some \(\theta<1/2\). Let \(S^{\mathrm{poly}}\) denotes the subclass of \(S\) of the functions with polynomial Euler product expressions \N\[\NF(s)=\prod_{p}\prod_{j=1}^{d}\left(1-\frac{\alpha_{j}(p)}{p^{s}}\right)^{-1}.\N\]\NLet \(z\) be a complex number. For \(\Im(z)>0\) and \(F\in{S^{\mathrm{poly}}}\), let \N\[\Nf(z,F)=\lim_{n\to\infty}\sum_{\rho; 0\Im\rho<T_{n}}\frac{e^{\rho z}\zeta(\rho-1)}{F'(z)}.\N\]\NIn this paper under review, the author first proves that the limit exists for all \(z\) such that \(\Im(z)>0\). Second, in his main result (Theorem 4.2), he proves that for all \(F\in{S^{\mathrm{poly}}}\) with \((r,\lambda_{j})=(1,1)\) for all \(j\) and \(0\leq\mu=\mu_{1}<1\), the function \(f(z,F)\) has a meromorphic continuation to \(\Im (z)=y>-\pi\). A functional equation satisfies by \(f(z,F)\) is proved under some further conditions (See Theorem 4.4).