the theta3 in the Merlin transform yielding the zeta function is a half integral weight modular form there's a path in SL(2,R) from its untransformed to it's transformed version, so to complete it with another factor, and emplace that in the Mellin transform
and then ask: "how do the nontrivial roots of the zeta function change in process?" and of course, immediately we should be thinking about Iwasawa theory here.