  Our goal in this paper is to present a generalization of the spectral zeta
regularization for general Feynman amplitudes. Our method uses complex powers
of elliptic operators but involves several complex parameters in the spirit of
the analytic renormalization by Speer, to build mathematical foundations for
the renormalization of perturbative interacting quantum field theories. Our
main result shows that spectrally regularized Feynman amplitudes admit an
analytic continuation as meromorphic germs with linear poles in the sense of
the works of Guo-Paycha and the second author. We also give an explicit
determination of the affine hyperplanes supporting the poles. Our proof relies
on suitable resolution of singularities of products of heat kernels to make
them smooth.
  As an application of the analytic continuation result, we use a universal
projection from meromorphic germs with linear poles on holomorphic germs to
construct renormalization maps which subtract singularities of Feynman
amplitudes of Euclidean fields. Our renormalization maps are shown to satisfy
consistency conditions previously introduced in the work of
Nikolov-Todorov-Stora in the case of flat space-times.
