Computation of Stark-Tamagawa units

Let $K$ be a totally real number field and let $l$denote an odd prime number. We design an algorithm which computes strong numerical evidence for the validity of the ``Equivariant Tamagawa Number Conjecture'' for the ${\mathbb{Q} [G]}$-equivariant motive $h^0(\mathrm{Spec}(L))$, where $L/K$ is a cyclic extension of degree $l$ and group $G$. This conjecture is a very deep refinement of the classical analytic class number formula. In the course of the algorithm, we compute a set of special units which must be considered as a generalization of the (conjecturally existing) Stark units associated to first order vanishing Dirichlet $L$-functions.