Functional equation for $GL(3)\times GL(2)\times GL(1)$ L-functions

Question

For two Maass forms $$f(z)=\sum_{n\neq 0}a(n)\sqrt{2\pi y}K_{v_1-\frac{1}{2}}(2\pi|n|y)e^{2\pi inx}$$ and$$g(z)=\sum_{\gamma\in U_2(\mathbb{Z})\backslash SL(2,\mathbb{Z})} \,\,\,\,\,\sum_{m=1}^{\infty}\,\,\sum_{n\neq 0}\frac{b(m,n)}{|mn|}W_{\text{Jacquet}}\left(\begin{pmatrix} |mn| & & \\ & m & \\ & & 1 \end{pmatrix}\begin{pmatrix}\gamma & \\ & 1\end{pmatrix}z\,,\, v_2,\,\psi_{1,\frac{n}{|n|}} \right)$$ for $SL(2,\mathbb{Z}), SL(3,\mathbb{Z})$, respectively (assume $f$ is to be even), we know the functional equation of the Rankin-Selberg $L$-function $$L_{f\times g}(s)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{a(n)b(m,n)}{(m^2n)^s} $$ where $a(n),b(m,n)$ are the coefficients in the Fourier-Whittaker expansions of $f$ and $g$ resp. This follows from the following integral representation $$\int_{SL(2,\mathbb{Z})\backslash\mathbb{H}}f(z).g\left(\begin{pmatrix}z&\\&1 \end{pmatrix}\right)|\det(z)|^{s-\frac{1}{2}}d^{*}z=L_{f\times g}(s)G_{(v_1,v_2)} $$ Do we know functional equation for the twisted series $$L_{f_{\chi}\times g}(s)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{\chi(n)a(n)b(m,n)}{(m^2n)^s} $$ where $\chi$ is say primitive mod $p$(prime) ?