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Let $G$ be a split reductive group over a number field $F$. We consider thecomputation of the inner product of two $K$-spherical pseudo Eisenstein seriesof $G$ supported in $[T,\mathcal{O}(1)]$ by means of residues, following aclassical approach initiated by Langlands. We show that only the singularitiesof the intertwining operators due to the poles of the completed Dedekind zetafunction $\Lambda_F$ contribute to the spectrum, while the singularities causedby the zeroes of $\Lambda_F$ do not contribute to any of the iterated residueswhich arise as a result of the necessary contour shifts.  In the companion paper [DMHO] we use this result to explicitly determine thespectral measure of $L^2(G(F)\backslashG(\mathbb{A}_F),\xi)^K_{[T,\mathcal{O}(1)]}$ by a comparison of the iteratedresidues with the residue distributions of [HO1].

Residue distributions, iterated residues, and the spherical automorphic spectrum