A conjecture on coincidence among the zeta functions associated with the space of binary cubic forms

About twenty-five years ago T. Shintani defined and studied four Dirichlet series whose coefficients are class numbers of integral binary cubic forms by using the theory of prehomogeneous vector spaces. In this paper, we give a conjecture that two of these Dirichlet series are essentially the same as the remaining two series. The conjecture is based on our calculation of the first two hundred nonzero coefficients of the four Dirichlet series, and is also consistent with the known functional equation and residues at poles. If the conjecture is true, we get simpler symmetric functional equations than previously known. Namely, if we take a certain linear combination of Shintani's Dirichlet series, that a single function is invariant under the variable change of s into 1 - s.