$\DeclareMathOperator\GL{GL} \DeclareMathOperator\SO{SO} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B}$Let $F$ be a number field and $V$ be a $(2n+1)$-dimensional quadratic space over $F$. Let $G=\SO(V)$ be the isometric group of $V$.
Consider the Siegel Eisenstein series $E(s,g)$ of $G$. Then at which points $s \in \mathbb{C}$, is it known that the residues of $E(s,g)$ are constant? It seems that $s=n$ is such point but I don't know why.
I appreciate your help on this!
