Let $K$ a totaly real number field, $\mathcal{O}\_K$ its ring of integre and $h$ the narrow class number of $K$. Let $\mathbf{f}$ a collection $(f\_1, ..., f\_h)$ of Hilbert cusp forms $f\_\lambda$
$(\lambda =1, ..., h)$ of weight $k=(k\_1,\dots,k\_n)$ with respect to $\Gamma\_{\lambda}(\mathcal{N}),$ $$R\_{\mathbf{f}}(s)=\sum\_{\mathfrak{m}\subseteq\mathcal{O}\_K}\frac{C(\mathfrak{m},\mathbf{f})^2}{\rm{N}(\mathfrak{m})^s}$$
the Rankin-Selberg zeta function, associated to $\mathbf{f}.$ can someone describe the analytic proprities (meromorphic continuation, functional equation, poles and residues) of $R\_{\mathbf{f}}(s).$



Poles of the Rankin-Selberg zeta function associated to Hilbert cusp forms
(Thanks again to GH-from-MO for explaining the convention in the relevant paper.) I think this is easier to understand from a (genuinely) adelic viewpoint, since that makes the proof(s) be the same as for the classic Rankin-Selberg story from 1939, in the same way that Iwasawa-Tate's viewpoint on Hecke L-functions makes Hecke's general case nearly identical to Riemann's original argument. One does not have to explicitly mention class numbers and so on.


Namely, as is in principle well-known, for cuspform $f$ (holomorphic or not) one integrates $|f|^2$ against a suitable Eisenstein series, whose meromorphic continuation and functional equation in the small case of $GL\_2$ come from Poisson summation. The "uneven" weights, at least if they are all of the same parity, are easily accommodated by fairly obvious choices of vectors in the principal series from which the Eisenstein series is formed. Since it's $|f|^2$, the parities are all "even", so there is no obstacle.


A genuine adelic set-up also presents all the Euler factors as local integrals against local Whittaker functions... whose computations are essentially identical to the classical case, at least at good primes. The archimedean integrals are the same.


