Let $p$ and $q$ be two distinct primes. Let $f\in \mathcal{S}\_k^{\ast}(pq,\psi)$ be a holomorphic newform of level $pq$, nebentypus $\psi$, and weight $k$, where $\psi = \chi\_p \chi\_{0(q)}$, with $\chi\_p$ a primitive character modulo $p$ and $\chi\_{0(q)}$ the principal character modulo $q$ (i.e. $\psi$ is an imprimitive character induced from the primitive character $\chi\_p$). Let $g$ be another holomorphic newform of level $q$, weight $k\_g$, and with trivial nebentypus.


Let $L(s,f \otimes g)$ be the Rankin-Selberg convolution $L$-function. Let $\Lambda(s,f\otimes g)=Q(f\otimes g)^{s/2} L\_\infty(s, f \otimes g)L(s, f \otimes g)$ be the complete $L$-function, where $Q(f\otimes g)$ is the conductor of $L(s, f\otimes g)$. Then we have the functional 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 are the root number $\epsilon(f\otimes g)$ and the conductor $Q(f\otimes g)$ in this case (the conductor equals $p^2 q^2$, I think)? Can anyone calculate the local $\epsilon\_v$-factor at the place $v \mid q$ 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.



Root number of the Rankin-Selberg convolution of two newforms
You need to do this via a local argument. A good reference for local components of $\mathrm{GL}\_2 \times \mathrm{GL}\_2$ automorphic representations is Gelbart and Jacquet, "[A relation between automorphic representations of $\mathrm{GL}(2)$ and $\mathrm{GL}(3)$](https://dx.doi.org/10.24033/asens.1355)". For just the $\mathrm{GL}\_2$ theory, see Schmidt, "[Some remarks on local newforms for $\mathrm{GL}(2)$](http://www2.math.ou.edu/~rschmidt/papers/gl2.pdf)".


* At all primes $v \nmid pq$, the local epsilon factors and conductor exponents are trivial.
* The local component of $f$ at $p$ is a principal series representation $\pi\_{f,p} = \omega\_{f,p,1} \boxplus \omega\_{f,p,2}$, where $\omega\_{f,p,1}, \omega\_{f,p,2}$ are character of $\mathbb{Q}\_p^{\times}$ of conductor exponent $c(\omega\_{f,p,1}) = 1$ and $c(\omega\_{f,p,2}) = 0$, while the local component of $g$ is a spherical principal series representation $\pi\_{g,p} = \omega\_{g,p,1} \boxplus \omega\_{g,p,2}$ with both characters unramified, so that $c(\omega\_{g,p,1}) = c(\omega\_{g,p,2}) = 0$. Then
\[\pi\_{f,p} \otimes \pi\_{g,p} = \omega\_{f,p,1} \omega\_{g,p,1} \boxplus \omega\_{f,p,1} \omega\_{g,p,2} \boxplus \omega\_{f,p,2} \omega\_{g,p,1} \boxplus \omega\_{f,p,2} \omega\_{g,p,2}.\]
The conductor exponent is
\[c(\pi\_{f,p} \otimes \pi\_{g,p}) = c(\omega\_{f,p,1} \omega\_{g,p,1}) + c(\omega\_{f,p,1} \omega\_{g,p,2}) + c(\omega\_{f,p,2} \omega\_{g,p,1}) + c(\omega\_{f,p,2} \omega\_{g,p,2}),\]
which is
\[1 + 1 + 0 + 0 = 2.\]
The epsilon factor $\epsilon\_p(s,\pi\_{f,p} \otimes \pi\_{g,p},\psi\_p)$ ostensibly depends on an additive character $\psi\_p$ of $\mathbb{Q}\_p$, which we may choose to be unramified (i.e. $c(\psi\_p) = 0$), though the global epsilon factor is independent of this. Anyway, $\epsilon\_p(s,\pi\_{f,p} \otimes \pi\_{g,p},\psi\_p)$ is equal to
\[\epsilon\_p(s,\omega\_{f,p,1} \omega\_{g,p,1},\psi\_p) \epsilon\_p(s,\omega\_{f,p,1} \omega\_{g,p,2},\psi\_p) \epsilon\_p(s,\omega\_{f,p,2} \omega\_{g,p,1},\psi\_p) \epsilon\_p(s,\omega\_{f,p,2} \omega\_{g,p,2},\psi\_p),\]
which is
\[\left(\omega\_{g,p,1}(p) \epsilon\_p(s,\omega\_{f,p,1},\psi\_p)\right) \cdot \left(\omega\_{g,p,2}(p) \epsilon\_p(s,\omega\_{f,p,1},\psi\_p)\right) \cdot 1 \cdot 1 = \epsilon\_p(s,\pi\_{f,p},\psi\_p)^2.\]
(See Proposition 1.4 of Gelbart and Jacquet and equations (4) and (6) of Schmidt and use the fact that $g$ has principal nebentypus means that the central character $\omega\_{\pi\_{g,p}} = \omega\_{g,p,1} \omega\_{g,p,2}$ of $\pi\_{g,p}$ is trivial, so that $\omega\_{g,p,1}(p) \omega\_{g,p,2}(p) = 1$.) As the conductor exponent is $2$,
\[\epsilon\_p(s,\pi\_{f,p} \otimes \pi\_{g,p},\psi\_p) = \epsilon\_p\left(\frac{1}{2},\pi\_{f,p} \otimes \pi\_{g,p},\psi\_p\right) p^{-2\left(\frac{1}{2} - s\right)}.\]
* The local components of $f$ and $g$ at $q$ are special representations $\pi\_{f,q} = \omega\_{f,q} \mathrm{St}\_q$ $\pi\_{g,q} = \omega\_{g,q} \mathrm{St}\_q$ with $\omega\_{f,q}, \omega\_{g,q}$ characters of $\mathbb{Q}\_q^{\times}$ that are unramified and either trivial or quadratic. The conductor exponent of $c(\pi\_{f,q} \otimes \pi\_{g,q})$ is $2$, as $\pi\_{f,q} \otimes \pi\_{g,q}$ is the isobaric sum of an unramified character of $\mathrm{GL}\_1(\mathbb{Q}\_p)$ and the Steinberg representation of $\mathrm{GL}\_3(\mathbb{Q}\_p)$, with the conductor exponent of the former being $0$ and the latter being $2$. By Proposition 1.4 of Gelbart and Jacquet and equation (11) of Schmidt, the epsilon factor $\epsilon\_q(s,\pi\_{f,q} \otimes \pi\_{g,q},\psi\_q)$ is
\[\epsilon\_q\left(s + \frac{1}{2},\omega\_{g,q} \omega\_{f,q} \mathrm{St}\_q, \psi\_q\right) \epsilon\_q\left(s - \frac{1}{2},\omega\_{g,q} \omega\_{f,q} \mathrm{St}\_q, \psi\_q\right) = \epsilon\_q(s,\omega\_{g,q} \omega\_{f,q} \mathrm{St}\_q, \psi\_q)^2.\]
One can further show that this is equal to
\[\epsilon\_q(s,\pi\_{g,q}, \psi\_q)^2 = \epsilon\_q(s,\pi\_{f,q}, \psi\_q)^2 = p^{-2(\frac{1}{2} - s)}\]
(the reference for this is Section 11.12 of Goldfeld and Hundley's book combined with Schmidt's paper).
* Finally, the local component of $f$ at $\infty$ is the discrete series representation $\pi\_{f,\infty} = D\_{k-1}$ of weight $k$, while $\pi\_{g,\infty} = D\_{k\_g - 1}$. Then $\pi\_{f,\infty} \otimes \pi\_{g,\infty} = D\_{|k - k\_g|} \boxplus D\_{k + k\_g}$, and so
\[\epsilon\_{\infty}(s, \pi\_{f,\infty} \otimes \pi\_{g,\infty},\psi\_{\infty}) = \epsilon\_{\infty}(s,D\_{|k - k\_g|},\psi\_{\infty}) \epsilon\_{\infty}(s,D\_{k + k\_g},\psi\_{\infty}),\]
which is
\[i^{|k - k\_g| + 1} i^{k + k\_g + 1} = (-1)^{\max\{k,k\_g\}+1}.\]
The best reference for this is Knapp, "[Local Langlands Correspondence: The Archimedean Case](https://www.math.stonybrook.edu/~aknapp/pdf-files/motives.pdf)".


So defining the global epsilon factor
\[\epsilon(s,\pi\_f \otimes \pi\_g) = \epsilon\_{\infty}(s, \pi\_{f,\infty} \otimes \pi\_{g,\infty},\psi\_{\infty}) \prod\_{p'} \epsilon\_{p'}(s, \pi\_{f,p'} \otimes \pi\_{g,p'},\psi\_{p'}),\]
where $p'$ runs over all primes, the functional equation for the completed Rankin-Selberg $L$-function $\Lambda(s, \pi\_f \otimes \pi\_g)$ is
\[\Lambda(s, \pi\_f \otimes \pi\_g) = \epsilon(s,\pi\_f \otimes \pi\_g) \Lambda(1 - s, \widetilde{\pi}\_f \otimes \widetilde{\pi}\_g).\]
By the above discussion,
\[\epsilon(s,\pi\_f \otimes \pi\_g) = (pq)^2 (-1)^{\max\{k,k\_g\}+1} \epsilon\_p\left(\frac{1}{2},\pi\_{f,p},\psi\_p\right)^2.\]
Finally, we note that $\epsilon\_p\left(\frac{1}{2},\pi\_{f,p},\psi\_p\right) = \eta\_f(p)$, the pseudo-eigenvalue of $f$ corresponding to the Atkin-Lehner operator $W\_p$; see equation (7.6) of Templier, "[Voronoï Summation for $\mathrm{GL}(2)$](http://www.math.cornell.edu/~templier/papers/voronoi-gl2.pdf)", and equation (7.8) shows that
\[\eta\_f(p) = \overline{\lambda\_f}(p) \tau(\chi\_p) p^{-1/2}\]
(note that $|\lambda\_f(p)| = 1$).


