On Shintani’s ray class invariant for totally real number fields

We introduce a ray class invariant X(C) for a totally real field, following Shintani’s work in the real quadratic case. We prove a factorization formula X(C) = X1(C) · · ·Xn(C) where each Xi(C) corresponds to a real place (Theorem 3.5). Although this factorization depends a priori on some choices (especially on a cone decomposition), we can show that it is actually independent of these choices (Theorem 4.9). Finally, we describe the behavior of Xi(C) when the signature of C at a real place is changed (Theorem 5.1). This last result is also interpreted into an interesting behavior of the derivative L′(0, χ) of L-functions.