I'm reading Kudla's Article on the Local Langlands Conjecture for $p$-adic general linear groups, and specifically I'm trying to understand how the ideas of Bernstein-Zelevinski yield show that you only need to prove the LLC for supercuspidal representations (on the automorphic side) and irreducible representatons (on the Galois side).


The part I can't find a reference on is: getting the $L$-functions to match. In particular, if $\tau\_1,\,\ldots\,\,\tau\_r$ are essentially-square-integrable representations, then they correspond to indecomposable representations $\rho\_1',\,\ldots,\,\rho\_r'$ on the Galois side. Then the correspondence gives
$$Q(\tau\_1,\ldots,\, \tau\_r) \mapsto \rho\_1' \oplus\ldots\oplus \rho\_r'$$
(here $Q$ is the Langlands quotient, i.e. the unique irreducible quotient of the representation parabolically induced from $\tau\_1\otimes \ldots\otimes \tau\_r$).
and so for match of $L$-functions we need
$$L(Q(\tau\_1,\ldots,\, \tau\_r),\,s) = \prod\_i L(\rho\_i',\,s) = \prod\_i L(\tau\_i,\,s).$$


The fact that
$$L(Q(\tau\_1,\ldots,\, \tau\_r),\,s) = \prod\_i L(\tau\_i,\,s)$$
Is stated in Kudla's article, but is not proved, and a reference is not provided. I was wondering if anyone could point me in the right direction, or explain why this is true?


(I guess the same question goes for $\epsilon$ factors but these are much more mysterious to me).


EDIT: unnecessary question removed, then edited for clarity.


Thanks!



L functions of Langlands Quotients of essentially-square-integrable representations
The identity $L(Q(\tau\_1,\ldots,\, \tau\_r),\,s) = \prod\_i L(\tau\_i,\,s)$ is part of Theorem 3.4 in Jacquet: Principal $L$-functions of the linear group, Proc. Symp. Pure Math. 33 (1979), Part 2, 63-86.


