Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $n$ be a positive integer. Let $(\pi,V)$ and $(\pi',V')$ be two irreducible generic Casselman-Wallach representations of $G_n=\operatorname{GL}_{n}(\mathbb{R})$ on Frechet spaces $V$ and $V'$ respectively.
Further, let $\mathcal{W}(\pi,\psi)$ and $\mathcal{W}(\pi',\psi^{-1})$ denote the Whittaker models of $\pi$ and $\pi'$ respectively. Finally, set $\mathcal{S}_n$ to be the space of Schwartz functions on $\mathbb{R}^n$.
Consider the following family of integrals: $$\mathcal{J}=\{I(s,W,W',\Phi):W\in\mathcal{W}(\pi,\psi),W'\in\mathcal{W}(\pi',\psi^{-1}),\Phi\in\mathcal{S}_n\}$$ defined by $$ \begin{aligned} &I(s, W,W', \Phi) \\ &=\int_{N_n \backslash G L_n} W\left(g\right)W'\left(g\right)\Phi(e_ng)|\operatorname{det}(g)|^{s} d g, \end{aligned} $$
$e_n=(0,0,...,0,1)\in\mathbb{R}^n.$
$N_n$ is the group of upper triangular matrices with unit diagonal over $\mathbb{R}.$
I have the following set of questions:
$(1)$ Is it possible that for every $s_0\in\mathbb{C}$, there exist $W\in\mathcal{W}(\pi,\psi),W'\in\mathcal{W}(\pi',\psi^{-1})$ and $\Phi\in\mathcal{S}_n$ such that $I(s, W,W', \Phi)$ is holomorphic and non-vanishing at $s_0$? I am not sure whether one can ensure holomorphy at $s_0$.
$(2)$ Is question $(1)$ known for Flicker and Jacquet Shalika integral representations?
Any help would be greatly appreciated.
