  Various types of local zeta functions studied in asymptotic group theory
admit two natural operations: (1) change the prime and (2) perform local base
extensions. Often, the effects of both of these operations can be expressed
simultaneously in terms of a single explicit formula. We show that in these
cases, the behaviour of local zeta functions under variation of the prime in a
set of density 1 in fact completely determines these functions for almost all
primes and, moreover, it also determines their behaviour under local base
extensions. We discuss applications to topological zeta functions, functional
equations, and questions of uniformity.
