Let $p$ and $M$$q$ be two distinct primes. Let $f\in H_k^*(pM,\psi)$$f\in \mathcal{S}_k^{\ast}(pq,\psi)$ be a holomorphic newform of level $pM$$pq$, nebentypus $\psi$, and weight $k$, where $\psi(mod\ pM)=\chi_p\chi_0$$\psi = \chi_p \chi_{0(q)}$, with $\chi_p$ a primitive character modulo $mod\ p$$p$ and $\chi_0$ a$\chi_{0(q)}$ the principal character modulo $mod\ M$$q$ (i.e., $\psi$ is an imprimtiveimprimitive character induced from the primitive character $\chi_p$). Let $g$ be another holomorphic newform, of level $M$$q$, weight $k_g$, and with trivial nebentypus.
Let $L(f\otimes g,s)$$L(s,f \otimes g)$ be the Rankin-Selberg convolution $L$-function. Let $\Lambda(f\otimes g,s)=Q(f\otimes g)^{s/2}L_\infty(f\otimes g)L(f\otimes g,s)$$\Lambda(s,f\otimes g)=Q(f\otimes g)^{s/2} L_\infty(s, f \otimes g)L(s, f \otimes g)$ be the complete $L$-function, where $Q(f\otimes g)$ is the conductor of $L(f\otimes g,s)$, then$L(s, f\otimes g)$. Then we have the functional equation \begin{equation}\Lambda(f\otimes g,s)=\epsilon(f\otimes g)\overline{\Lambda(f\otimes g,1-\bar{s})}.\end{equation}\begin{equation} \Lambda(s, f\otimes g)=\epsilon(f \otimes g) \overline{\Lambda(f\otimes g,1-\bar{s})}. \end{equation}
My question is: what
What are the root number $\epsilon(f\otimes g)$ and the conductor $Q(f\otimes g)$ in this case (the conductor equals $p^2M^2$$p^2 q^2$, I think)? Can anyone calculate the local $\epsilon_v$-factor at the place $v|M$$v \mid q$ for me? I know a good reference is http://tan.epfl.ch/files/content/sites/tan/files/PhMICHELfiles/RSfinal.pdf, but most of their statements are for the levels of $f$ and $g$ to be co-prime.
