Residue of Eisenstein Series on GL(n)

Reference: Mœglin, C.; Waldspurger, J.-L.: Le spectre residuel de GL(n)

On GL($n$), the result of Waldspurger shows that if an automorphic representation $\pi$ is non-cuspidal and in the discrete spectrum, then there is $d|n$ and an automorphic representation $\tau$ on GL($d$) such that $\pi$ is the residue of the Eisenstein seires $E(*,\tau,n/d)$ induced from a direct sum of ($n/d$) copies of $\tau$.

I am wondering what these $\pi$ look like? What's their $L$-functions (in terms of $\tau$)?

The $L$-function of $E(*,\tau,n/d)$ may look like $L(s,\tau)^{n/d}$? But how about its residue?