  We investigate the Shintani zeta functions associated to the prehomogeneous
spaces, the example under consideration is the set of $2 \times 2\times 2$
integer cubes. We show that there are three relative invariants under a certain
parabolic group action, they all have arithmetic nature and completely
determine the equivalence classes. We show that the associated Shintani zeta
function coincides with the $A_3$ Weyl group multiple Dirichlet series.
Finally, we show that the set of semi-stable integer orbits maps finitely and
surjectively to a certain moduli space.
