Suppose $P \subseteq GL(n)$ is a parabolic subgroup. How do I find (or what is a reference for) the poles and residues of degenerate Eisenstein series associated to $P$?

More precisely, suppose $P_1, \ldots, P_r$ are parabolic subgroups of $GL(n)$ containing $P$, $\delta_{P_i}$ denotes the modulus character of $P_i$, $s_1, \ldots, s_r$ are complex parameters, and $f(g, s_1, \ldots, s_r) \in Ind_{P}^{GL(n)}(\delta_{P_1}^{s_1} \cdots \delta_{P_r}^{s_r})$. The degenerate Eisenstein series associated to this data is

$E(g,f,s_1, \ldots, s_r) = \sum_{\gamma \in P(F)\backslash GL_n(F)}{f(\gamma g, s_1, \ldots, s_r)}.$

Where are the poles of this Eisenstein series located, and what are the residues at these poles?

If $P$ is the maximal parabolic of $GL(n)$ that stabilizes a line in the $n$-dimensional representation of $GL(n)$, the above question may be answered using Poisson summation. Outside of this special case, any comments or references would be helpful.

Suppose $P \subseteq GL(n)$ is a parabolic subgroup. How do I find (or what is a reference for) the poles and residues of degenerate Eisenstein series associated to $P$?

More precisely, suppose $P_1, \ldots, P_r$ are parabolic subgroups of $GL(n)$ containing $P$, $\delta_{P_i}$ denotes the modulus character of $P_i$, $s_1, \ldots, s_r$ are complex parameters, and $$f(g, s_1, \ldots, s_r) \in Ind_{P}^{GL(n)}(\delta_{P_1}^{s_1} \cdots \delta_{P_r}^{s_r}).$$ The degenerate Eisenstein series associated to this data is $$E(g,f,s_1, \ldots, s_r) = \sum_{\gamma \in P(F)\backslash GL_n(F)}{f(\gamma g, s_1, \ldots, s_r)}.$$

Question. Where are the poles of this Eisenstein series located, and what are the residues at these poles?

If $P$ is the maximal parabolic of $GL(n)$ that stabilizes a line in the $n$-dimensional representation of $GL(n)$, the above question may be answered using Poisson summation. Outside of this special case, any comments or references would be helpful.