Signed Shintani cones for number fields with one complex place

We give a signed fundamental domain for the action on $$\mathbb{C}$^*\times $\mathbb{R}$+^{n-2}$ofthetotallypositiveunits$E(k)+$ofanumberfield$k$ofdegree$n$ and having exactly one pair of complex embeddings. This signed fundamental domain, built of $k$-rational simplicial cones, is as convenient as a true fundamental domain for the purpose of studying Dedekind zeta functions. However, while there is no general construction of a true fundamental domain, we construct a signed fundamental domain from any set of fundamental units of $k$.