Functional equation of Rankin-Selberg zeta function

Question

Can someone explain to me where come from the functional equation (1.6) in this article ?

Answer 1

The short answer is that the functional equation appears in the cited work of Rankin and Selberg (independently).

For a slightly longer (classical) answer, suppose $f(z)$ is the weight $12$ cusp form on $\text{SL}(2, \mathbb{Z})$ whose coefficients are the Ramanujan $\tau$ function. And let $E(z,s)$ be Selberg's Eisenstein series $$ \zeta(2s) \sum_{\gamma \in \Gamma_\infty \backslash \text{SL}(2,\mathbb{Z})} \text{Im} (\gamma z)^s. $$ Then the functional equation $s \mapsto 1-s$ of the Eisenstein series is a classical computation (which is easily searchable). Computing the Petersson inner product gives the Rankin-Selberg zeta function associated to $f$ in the sense that $$ \langle \lvert f \rvert^2 y^{12}, E(\cdot,s)\rangle = \frac{\Gamma(s + 11)}{(4\pi)^{s + 11}} \sum_{n \geq 1} \frac{\tau(n)^2}{n^{s+11}}. $$ In this form, the functional equation of the Eisenstein series gives exactly the functional equation of the Rankin-Selberg zeta function.

For more on Eisenstein series, I would recommend looking at Diamond and Shurmans A first course in modular forms. For more on Selberg's Eisenstein series, I recommend looking at Goldfeld's Automorphic forms on $\text{GL}(n, \mathbb{R})$, or even at the original papers of Ranking and Selberg. Each of these includes a discussion of the Rankin-Selberg convolution and zeta function.

Answer 2

You can check Cogdell's notes, say https://people.math.osu.edu/cogdell.1/ictp-www.pdf

For your case, you also need to calculate the local gamma factor at the infinity place, which is quite standard.