I understand that there is a functional equation for ideal class zeta functions of number fields. That is, if $k$ is a number field and $C$ is a class of fractional ideals of $k$, then
$\zeta_C(s) = \sum_{\substack{\mathfrak{a} \in C\\ \mathfrak{a} \text{ integral}}} \frac{1}{N(\mathfrak{a})^s}$
has a functional equation relating $\zeta_C(s)$ to $\zeta_C(s-1)$ (I guess this is in Neukirch, Algebraic Number Theory, Chapter VII). Does such a functional equation exist for general ray class zeta functions, where $C$ is replaced by a ray class modulo some modulus? I am especially interested in when the modulus includes some infinite places, i.e., for "narrow ideal class zeta functions".
(I guess that maybe the answer is yes, since Hurwitz zeta functions satisfy a functional equation, at least for a limited range of $s$.)
As @user1952009 commented, yes, with characters with the same conductor (and maybe same sign/epsilon factor in functional equation), linear combinations obviously inherit the functional equation.
For example, for not-narrow ideal classes, all the Hecke L-functions have the same functional equation, so arbitrary linear combinations inherit it, and the "ideal class zeta" you wrote is such a linear combination (by the usual Fourier inversion on the finite abelian group that is the class group).
A superficial thought experiment makes me think that the narrow-class-group characters all give L-functions with the same functional equation, too.
For general ray-class things not all characters will give L-functions with the same functional equation, so $\zeta_C(1-s)$ will be a messier linear combination of $\Lambda(1-s,\chi)$'s.