In this paper, we present formulas for the edge zeta function and the second weighted zeta function with respect to the group matrix of a finite abelian group $Γ$. Furthermore, we give another proof of Dedekind Theorem for the group determinant of $Γ$ by the decomposition formula for a matrix of a group covering of a digraph. Finally, we treat the weighted complexity of the complete graph with entries of the group matrix of $Γ$ as arc weights.