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We study the class numbers of integral binary cubic forms. For each $SL_2(Z)$invariant lattice $L$, Shintani introduced Dirichlet series whose coefficientsare the class numbers of binary cubic forms in $L$. We classify the invariantlattices, and investigate explicit relationships between Dirichlet seriesassociated with those lattices. We also study the analytic properties of theDirichlet series, and rewrite the functional equation in a self dual form usingthe explicit relationship.

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