  In the present paper, we treat multidimensional polynomial Euler products
with complex coefficients on ${\mathbb{R}}^d$. We give necessary and sufficient
conditions for the multidimensional polynomial Euler products to generate
infinitely divisible, quasi-infinitely divisible but non-infinitely divisible
or not even characteristic functions by using Baker's theorem. Moreover, we
give many examples of zeta distributions on ${\mathbb{R}}^d$ generated by the
multidimensional polynomial Euler products with complex coefficients. Finally,
we consider applications to analytic number theory.
