Let $\Gamma = \operatorname{SL}_2(\mathbb{Z})$ and $\mathcal{H}$ be the upper half-plane. For $A>1$, define the truncated Eisenstein series $E_A(z,s)$ as $$E_A(z,s) = \begin{cases} E(z,s), & \text{if } z \in \mathcal{F}_A, \ E(z,s) - e(y,s), & \text{if } z \in \mathcal{C}_A, \end{cases}$$ where $\mathcal{F}_A = {z \in \mathcal{F} : \Im(z) \leq A}$, $\mathcal{C}_A = {z \in \mathcal{F} : \Im(z) > A}$, $\mathcal{F}$ is the standard fundamental domain for $\Gamma \backslash \mathcal{H}$, and $e(y,s)$ is the constant term in the Fourier expansion of $E(z,s)$.
Consider the following integral transform:
$$V(x,t) = \frac{1}{2\pi i} \int_{(\sigma)} H(s,t) (\pi A x)^{-s} \frac{ds}{s},$$
where $\sigma > 0$ is fixed, $x>0$, and
$$H(s,t) = \frac{\prod_{\pm} \Gamma(\frac{s}{2} + \frac{1}{4} + iT \pm it) \Gamma(\frac{s}{2} + \frac{1}{4} - iT \pm it)}{\Gamma(s + \frac{1}{2} + iT) \Gamma(\frac{1}{2} + iT) \Gamma(\frac{1}{2} + it)}.$$
What is the asymptotic behavior of $V(x,t)$ as $T \rightarrow \infty$ where $T \rightarrow \infty$ and $T^{1-\alpha} < |t| < 2T - T^{1-\alpha}$ for a fixed small $\alpha > 0$?
