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Let $f$ be a Maass cusp form for $\text{SL}_2(\mathbb{Z})$ on the upper half plane. Let $\varphi = \varphi_f$ be its lift to an automorphic form on $G = \text{PGL}_2(\mathbb{R})$ and let $\pi = \pi_{\varphi}$ denote the cyclic subrepresentation of $L^2(\Gamma \backslash G)$ (where $\Gamma = \text{PGL}_2(\mathbb{Z})$) generated by $\varphi$. 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).

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).

Let $f$ be a Maass cusp form for $\text{SL}_2(\mathbb{Z})$ on the upper half plane. Let $\varphi = \varphi_f$ be its lift to an automorphic form on $G = \text{PGL}_2(\mathbb{R})$ and let $\pi = \pi_{\varphi}$ denote the cyclic subrepresentation of $L^2(\Gamma \backslash G)$ (where $\Gamma = \text{PGL}_2(\mathbb{Z})$) generated by $\varphi$. 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 $\{|\Re(s)-1/2| \geq A\}$ for some $A > 0$ (I think $A = 2.5$ is valid).

Let $f$ be a Maass cusp form for $\text{SL}_2(\mathbb{Z})$ on the upper half plane. Let $\varphi = \varphi_f$ be its lift to an automorphic form on $G = \text{PGL}_2(\mathbb{R})$ and let $\pi = \pi_{\varphi}$ denote the cyclic subrepresentation of $L^2(\Gamma \backslash G)$ (where $\Gamma = \text{PGL}_2(\mathbb{Z})$) generated by $\varphi$. 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).

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

$$W_{\varphi}(g) = \int_{\mathbb{Z} \backslash \mathbb{R}}{\varphi(n(x)) e(-x)}dx\qquad n(x) = \begin{pmatrix} 1 & x\\ 0 & 1 \end{pmatrix} $$ 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 continuity for compact subsets of the plane excluding a vertical strip around the half-line.

Let $f$ be a Maass cusp form for $\text{SL}_2(\mathbb{Z})$ on the upper half plane. Let $\varphi = \varphi_f$ be its lift to an automorphic form on $G = \text{PGL}_2(\mathbb{R})$ and let $\pi = \pi_{\varphi}$ denote the cyclic subrepresentation of $L^2(\Gamma \backslash G)$ (where $\Gamma = \text{PGL}_2(\mathbb{Z})$) generated by $\varphi$. 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 $\{|\Re(s)-1/2| \geq A\}$ for some $A > 0$ (I think $A = 2.5$ is valid).