  Inspired by the invariant of a number field given by its zeta function, we
define the notion of {\it weak arithmetic equivalence}, and show that under
certain ramification hypothesis, this equivalence determines the local root
numbers of the number field. This is analogous to a result of Rohrlich on the
local root numbers of a rational elliptic curve. Additionally, we prove that
for tame non-totally real number fields, the integral trace form is invariant
under weak arithmetic equivalence.
