Let $f \in \mathscr{S}_{k}(N,\chi)$ and $g \in \mathscr{S}_{k}(M,\psi)$ be newforms with $(N,M) = 1$. For $\Re(s) > 1$, I was able to derive an integral representation for the Rankin-Selberg convolution: $$L(s,f \otimes g) = \frac{(4\pi)^{s+k-1}\pi^{s}}{\Gamma(s+k-1)\Gamma(s)}\int_{\mathscr{F}}f(z)\overline{g(z)}\Im(z)^{k}\frac{L(2s,\chi\psi)}{\zeta(2s)}E^{\ast}(z,s)\,d\mu.$$ where $E^{\ast}(z,s)$ is the completed real-analytic Eisenstein series. I'm trying to use this expression to derive a functional equation for $L(s,f \otimes g)$ but substiuting in the functional equations for the Dirichlet L-series, zeta function, and real-analytic Eisenstein series doesn't seem to give me what I want. The gamma factors don't symmetrize nicely as $s \to 1-s$ and I dont get the Fricke involutions of $f$ and $g$ to appear (I think they should appear because they give the dual $L$-functions for $L(s,f)$ and $L(s,g)$). Can anyone tell me what the functional equation should be and if I've made an error with this integral representation?
I was able to have sucess when I did all this on level $1$ ($N = M = 1$), since the Dirichlet $L$-series and zeta functions don't appear and thus the integrand is invariant as $s \to 1-s$ but I'm having trouble extending to the case with fixed weight but different levels.