A unusual Eisenstein type series

Question

When working on certain integral transforms I have come across the following unusual looking series;

$$ E(z_1,z_2,s) = \sum_{ \{ \ (a,b,c,d)\in \mathbb{Z}^4 \ | \ ad-bc=1 \ \} } \frac{ \Im(z_1)^s\Im(z_2)^s}{ |(az_1+b)\bar{z}_2+(cz_1+d)|^{2s}} $$ where $\Im(z_1),\Im(z_2)>0$, and $\Re(s)>>0$.

I would be very interested i knowing whether this function is known to have a meromorphic continuation in $s$, and - assuming such a meromorphic continuation exists - whether there exist a way of evaluating this function to high precision outside the half-plane where the above series converges.

More generally I would be interested in knowing whether this function is known in the literature, and if so under which name.

UPDATE: Let me say a few words about what I would like to do with these functions. Setting $$ E^*(z_1,z_2,s) = \Gamma(s) E(z_1,z_2,s)$$ my goal would be to make sense of $E^*(z_1,z_2,-n)$ where $n$ is any positive integer, preferably by expressing it as a sum of simpler/more well-known functions. Ideally this would enable me to make sense of the integral $$ R.N. \int_0^{\infty} E^*(z_1,iy,-n) y^m \frac{dy}{y}$$

A quick calculation where I completely ignore any question of convergence seems to indicate that the integral given above might be expressible as a finite sum of functions of the form $$P_k(z,s) = \sum_{n\in \mathbb{Z}} \sum_{\gamma} \frac{\Im(\gamma.z)^s}{n^k} e^{2\pi i n \gamma.z}$$ where we are summing over $\gamma\in \operatorname{SL}_2(\mathbb{Z})$. This is done by rewriting the original sum defining $E(z_1,z_2,s)$ as a sum over $K$-Bessel functions, and then using that $K_{n-1/2}$ is elementary. Sadly I can't seem to turn this line of reasoning into anything more than a purely formal argument, hence it may very well be completely nonsensical. My hope was that either the function $E(z_1,z_2,s)$ was known in the literature and that people had studied it and related it to other more well-known objects, or that there existed a rapidly converging expression allowing me to at least numerically test the extent to which the above formal argument might have some merit.

I don't know whether my updated remark shows anything other than my own blind optimism when it comes to formal arguments. Let me close by emphsizing that I would be very interested in any thoughts on the original question, regardless of whether they may or may not relate to the updated remark.

Answer 1

It may seem tangential, but it is encouraging: if the sum were over all integer matrices with positive integer determinant, this would be the restriction to $SL_2(\mathbb Z)\times SL_2(\mathbb Z)$ of a (waveform) Siegel-type Eisenstein series from $Sp_4$, with a product of $SL_2(\mathbb Z)$ Eisenstein series $E_s\otimes E_s$ subtracted. Thus, this modified form certainly has a meromorphic continuation, although the question of its behavior off to the left is non-trivial.

The "smaller" sum over integer matrices with determinant $1$ probably needs to be spectrally decomposed (as a right $K=SO(2)$-invariant $L^2$ function on $SL_2(\mathbb Z)\backslash SL_2(\mathbb R)$). Since the function $\varphi(z_1,z_2)=y_1^s y_2^s/|z_1\overline{z}_2+1|^{2s}$ has the property $\varphi(gz_1,g^\tau z_2)=\varphi(z_1,z_2)$ with $g^\tau=(g^\top)^{-1}$, for $g\in SL_2(\mathbb R)$, not merely for $g\in SL_2(\mathbb Z)$, this situation falls into examples considered already by Selberg as "point-pair invariants", although there is potential for analytical trouble due to the non-compact support.

It is easier to execute some formalities on the group, rather than on the domain, and this also makes visible certain potential "trick-applications" of some simple representation theory. So write $\varphi_w(g,h)=\Im(g.i)^w\,\Im(h.i)^w/|(g.i)\overline{(h.i)}+1|^{2w}$, and $\Phi_w(g,h)=\sum_\gamma \varphi(\gamma.g,h)$. For eigencuspform (waveform) $f$, the $f$-component of $\Phi_w(-,h)$ is computed first by unwinding $$ \int_{\Gamma\backslash G} \Phi_w(g,h)\;\overline{f}(g)\;dg \;=\; \int_G \varphi_w(g,h)\;\overline{f}(g)\;dg $$ and then by a change of variables, after using the point-pair property: $$ \ldots\;=\; \int_G \varphi_w(h^\tau g,1)\;\overline{f}(g)\;dg \;=\; \int_G \varphi_w(g)\;\overline{f}(h^\top\cdot g)\;dg $$ This presents the $f$-component (with fixed $h$) as an integral operator on the representation space in which $f$ lies. Since $f$ is an eigenform-waveform, it is the essentially unique spherical vector in the (copy of) principal series repn it generates. Since this integral operator preserves the spherical-ness, it multiplies the spherical vector by a scalar depending only upon $w$ and upon the parameter describing the principal series.

Thus, this scalar can be computed on any model of the principal series. Rather than looking at Whittaker models, by the nature of the integral kernel here we can just use the standard model, which has spherical function $z\rightarrow y^s$. The indicated integral is (something like...) $$ {\Gamma(w-{1\over 2}) \cdot \Gamma(s+w-1)\cdot \Gamma(w-s) \over \Gamma(w)\cdot \Gamma(2w-1)} $$ The spectral expansion converges in $L^2$ at least for $Re(w)\gg 1$. Thus, the discrete-spectrum part has a meromorphic continuation (in $L^2$ at least) apart from poles at $w=s_j$ with spectral parameters $s_j$ of cuspforms and the constant function.

The continuous-spectrum part seems ok, too, but its analytic behavior to the left of $\Re(w)=1/2$ is a little trickier: in the strip $-\ell-{1\over 2}<\Re(w)<-\ell+{1\over 2}$ (with non-negative integer $\ell$) there are additional terms $E_{w+\ell}\otimes E_{1-w+\ell}$ multiplied by the vertical integral on $\Re(s)=1/2$ of that expression in Gamma-functions above... Details to track, if this does matter.

The questioner's proposed multiplication by (in effect) $\Gamma(w)$ would cancel the $\Gamma(w)$ in the denominator observable in the expression above, but the speculated-upon Mellin transform would, in effect, be evaluating the standard $L$-function of waveforms, weighting them by that Gamma expression above, and adding them. The continuous spectrum part would be a finite sum of integrals of zetas.

I've not attempted to keep track of all details in this sketch, not knowing whether this is going in a desirable direction or not. If this seems purposeful, let me know and maybe I'll make and link-to a PDF.