 Zeta-completed Rankin-Selberg coefficient series

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 $k$.