Relations among Dirichlet series whose coefficients are class numbers of binary cubic forms

We study the zeta functions for the space of binary cubic forms introduced by Shintani. The zeta function is defined for each invariant lattice. We classify the invariant lattices, and investigate explicit relationships between the zeta functions associated with those lattices. We also study the analytic properties of the zeta functions, and rewrite Shintani's functional equation in a self dual form using an explicit relation.