  Given a number field $K$ with a Hecke character $\chi$, for each place $\nu$
we study the free scalar field theory whose kinetic term is given by the
regularized Vladimirov derivative associated to the local component of $\chi$.
These theories appear in the study of $p$-adic string theory and $p$-adic
AdS/CFT correspondence. We prove a formula for the regularized Vladimirov
derivative in terms of the Fourier conjugate of the local component of $\chi$.
We find that the Green's function is given by the local functional equation for
Zeta integrals. Furthermore, considering all places $\nu$, the field theory
two-point functions corresponding to the Green's functions satisfy an adelic
product formula, which is equivalent to the global functional equation for Zeta
integrals. In particular, this points out a role of Tate's thesis in adelic
physics.
