= Zeta-completed Rankin-Selberg coefficient series
{title2=$\mathcal R(s)=2\pi^{-w}\Gamma(w)\zeta(2w)(4\pi)^{-s}\Gamma(s)F(s),\quad w=s-k+1$}
For two level-one weight-$k$ <cusp forms>, $F(s)=\sum a_n\overline b_n n^{-s}$. Unfolding against the nonholomorphic lattice sum normalized by $\pi^{-w}\Gamma(w)$ gives the displayed completion as the integral of $f\overline g y^k$ against that <Eisenstein series>. Its <functional equation> is $\mathcal R(s)=\mathcal R(2k-1-s)$, and its only possible simple poles are at $k-1,k$. The raw series is obtained by dividing out the gamma and zeta factors, so it does not inherit the same unqualified pole statement; zeros of the denominator zeta factor can yield additional possible poles. The raw series is regular at $k-1$ and can have a simple pole at $k$.
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