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.
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
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