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We give a signed fundamental domain for the action on $\mathbb{C}^*\times\mathbb{R}_+^{n-2}$ of the totally positive units $E(k)_+$ of a number field$k$ of degree $n$ and having exactly one pair of complex embeddings. Thissigned fundamental domain, built of $k$-rational simplicial cones, is asconvenient as a true fundamental domain for the purpose of studying Dedekindzeta functions. However, while there is no general construction of a truefundamental domain, we construct a signed fundamental domain from any set offundamental units of $k$.

Signed Shintani cones for number fields with one complex place