  Let $\mathfrak{Var}_k^G$ denote the category of pairs $(X,\sigma)$, where $X$
is a variety over $k$ and $\sigma$ is a group action on $X$. We define the
Grothendieck ring for varieties with group actions as the free abelian group of
isomorphism classes in the category $\mathfrak{Var}_k^G$ modulo a cutting and
pasting relation. The multiplication in this ring is defined by the fiber
product of varieties. This allows for motivic zeta-functions for varieties with
group actions to be defined. This is a formal power series
$\sum_{n=0}^{\infty}[\text{Sym}^n (X,\sigma)]t^n$ with coefficients in the
Grothendieck ring. The main result of this paper asserts that the motivic
zeta-function for an algebraic curve with a finite abelian group action is
rational. This is a partial generalization of Weil's First Conjecture.
