Let $p$ and $M$ be two distinct primes. Let $f\in H_k^*(pM,\psi)$ be a holomorphic newform of level $pM$, nebentypus $\psi$, and weight $k$, where $\psi(mod\ pM)=\chi_p\chi_0$, with $\chi_p$ a primitive character $mod\ p$ and $\chi_0$ a principal character $mod\ M$ (i.e., $\psi$ is an imprimtive character induced from the primitive $\chi_p$). Let $g$ be another holomorphic newform, of level $M$, weight $k_g$, and with trivial nebentypus. Let $L(f\otimes g,s)$ be the Rankin-Selberg convolution $L$-function. Let $\Lambda(f\otimes g,s)=Q(f\otimes g)^{s/2}L_\infty(f\otimes g)L(f\otimes g,s)$ be the complete $L$-function, where $Q(f\otimes g)$ is the conductor of $L(f\otimes g,s)$, 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}

My question is:
what are the root number $\epsilon(f\otimes g)$ and the conductor $Q(f\otimes g)$ in this case (the conductor equals $p^2M^2$, I think)? Can anyone calculate the local $\epsilon_v$-factor at the place $v|M$ 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.