Let $F$ be a p-adic field and $(\rho,V)$ be an n-dimensional indecomposable representation of the local Weil-Deligne group $W_F'$. Then we know that $\rho\simeq \rho'\otimes Sp(m)$, where $\rho'$ is an irreducible representation of $W_F$, and $Sp(m)=((\tau,N), W)$, where $W=\mathbb{Q}e_0\oplus\cdots\oplus\mathbb{Q}e_{m-1}$, and $\tau$ is a representation of $W_F$ given by $\tau(w)e_j=\vert\vert w\vert\vert^je_j$ and $Ne_j=e_{j+1}$ for all $0\leq j\leq m-1$, and $Ne_{m-1}=0$. Now I have a few questions about the local $\gamma$-factors:
(1) On the Artin side, does it follow that the local $\gamma$-factors satisfy that$$\gamma(s, \rho,\psi)=\gamma(s, \rho',\psi)?$$ Now if $r$ is an analytic representation of $GL_n(\mathbb{C})$, if we let $r'=r\vert_{V'\otimes 1}$, then is there any relationship between $\gamma(s, r\circ\rho,\psi)$ and $\gamma(s, r'\circ\rho',\psi)$?
(2) If $\rho'\leftrightarrow \pi(\rho')$ under LLC, it is then expected that we have $$\gamma(s, r'\circ\rho',\psi)=\gamma(s,\pi(\rho'),\psi),$$ suppose this holds, and we know that by Bernstein-Zelevinsky's theorem there exists a unique irreducible quotient $Q(\Delta)$ and a unique irreducible subrepresentation $Z(\Delta)$ of the induced representation $Ind_{GL_{n'}(F)^m}^{GL_n(F)}\pi'\otimes\pi'(1)\otimes\cdots\otimes\pi'(m-1)$, so is there any relationship between $\gamma(s, Q(\Delta), r,\psi)$, $\gamma(s, Z(\Delta),r,\psi)$ and $\gamma(s, \pi(\rho'), r',\psi)$ as long as these analytic local $\gamma$-factors are well-defined(e.g through Langlands-Shahidi method)?