Stark's conjecture over the rationals for Artin $L$-values

Let $K/k$ be a finite abelian extension of number fields with Galois group $G$, let $S$ be a finite set of places of $k$, and let $S_K$ be the places of $K$ above $S$. Let $\mathcal{A}$ be an Artin system of $S_K$-units. For $\chi\rd in \widehat{G}$, define

$$A(\chi,\mathcal{A})=\frac{L_{K,S}^{*}(0,\chi)}{R(\chi,\mathcal{A})}.$$

Stark's conjecture over the rationals. For every $\chi\in\widehat{G}$, one has $A(\chi,\mathcal{A})\in\overline{\mathbb{Q}}$ and

$$A(\chi,\mathcal{A})^{g}=A(\chi^{g},\mathcal{A})$$

for all $g\in\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. This is the rationality and Galois-equivariance prediction for the normalized leading terms of abelian Artin $L$-functions, and is used to place the associated group-ring element in $\mathbb{Q}[G]$.