Let $f$ be a Maass cusp form for $\text{SL}\_2(\mathbb{Z})$ on the upper half plane. Let $\varphi\_0$ be its lift to an automorphic form on $G = \text{PGL}\_2(\mathbb{R})$ and let $\pi = \pi\_{f} =\langle \varphi\_0 \rangle$ denote the cyclic subrepresentation of $L^2(\Gamma \backslash G)$ (where $\Gamma = \text{PGL}\_2(\mathbb{Z})$) generated by $\varphi\_0$. To each smooth vector $\varphi \in \pi$ we can associate its Whittaker function 


$$W\_{\varphi}(g) = \int\_{\mathbb{Z} \backslash \mathbb{R}}{\varphi(n(x)g) e(-x)}dx\qquad n(x) = \begin{pmatrix} 1 & x\\ 0 & 1 \end{pmatrix}
$$
and the Mellin-transform thereof, known as the local zeta integral attached to 
$\varphi$ : 
$$
Z(W\_{\varphi}, s) = \int\_{\mathbb{R}^{\times}}{W\_{\varphi}(a(y)) |y|^{s-1/2} d^{\times}y} \qquad a(y) = \begin{pmatrix} y & 0\\ 0 & 1 \end{pmatrix}
$$ 


It is known (in my case, from Proposition 1.5 in Cogdell Piatetski-Shapiro "The arithmetic and spectral analysis of Kloosterman sums") that the assignment $\varphi \mapsto Z(W\_{\varphi},s)$ is continuous for the smooth topology on $\pi^{\infty}$ and even uniformly so for $s$ varying in a compact subset of some right half-plane. 


Does this uniform continuity remain true for the meromorphic continuation of the local local zeta integral? In particular, I'd like to know whether this holds for compact subsets of $\Re(s) = 1/2$.


The local functional equation gives uniform continuity for $s$ varying in compact subsets of $\{|\Re(s)-1/2| > A\} - \{\text{poles}\}$ for some $A > 0$ (I think $A = 5/2$ is valid).



Meromorphic continuation of local zeta integrals
As we are considering continuity of the local zeta integral it is sufficient to pose the problem locally. Thus we think the Archimedean local component of $\pi$ is the principal series $\pi\_\mu$ where $\mu\in \mathbb{C}$ such that the Laplacian eigenvalue of $f$ is $1/4-\mu^2$.


Let $\{\mathcal{D}\_i\}$ be the set of invariant differential operators of degree $\le 2$. We define a Sobolev norm (semi-norm) on $\pi\_\mu$ by 
$$S(v):=\sum\_{i}\|\mathcal{D}\_iv\|\_\pi.$$
Thus to prove the uniform continuity it is enough to show the following: For every compact set $K\subseteq \{s\mid\Re(s)> 0\}$, 
$$|Z(W\_{v\_1},s)-Z(W\_{v\_2},s)|\ll\_K S(v\_1-v\_2),$$
whenever $s\in K$.
Using linearity of the zeta integral it is enough to show that 
$$Z(W\_v,s)\ll\_K S(v).$$
But the LHS is
$$\int\_{\mathbb{R}^\times}W\_v(a(y))|y|^{s-1/2}d^\times y\ll\_KS(v)\int\_{\mathbb{R}^\times}\min(|y|^{1/2-\eta},|y|^{-2})|y|^{\Re(s)-1/2}\frac{dy}{|y|}.$$
We have used Proposition 3.2.3 of [Michel-Venkatesh](https://link.springer.com/content/pdf/10.1007%2Fs10240-010-0025-8.pdf) in a slightly modified form (their seminorm is only on the Kirillov model of $\pi\_\mu$). If $\pi\_\mu$ is tempered, i.e. $\mu$ is purely imaginary then $\eta$ can be replaced by any $\epsilon > 0$ (with $\ll\_K$ replaced by $\ll\_{K,\epsilon}$). As $\Re(s)> 0$, the last integral is $\ll\_K 1$ choosing $\epsilon$ small enough.


