A functional equation between the $\zeta $ distributions can be obtained from the theory of prehomogeneous vector spaces. We show that the functional equation can be extended from the Schwartz space to certain degenerate principal series.
"A functional equation and degenerate principal series." Rocky Mountain J. Math. 46 (6) 1987 - 2016, 2016. https://doi.org/10.1216/RMJ-2016-46-6-1987