Stark's special-value conjecture for Artin representations

Let $\varrho$ be an Artin representation with coefficient field $E$, Artin conductor $f_\varrho$, completed $L$-function $\Lambda(s,\varrho)$, root number $W(\varrho)$, and Stark regulator matrix $R(\varrho)$. Let $a$ and $b$ be the archimedean gamma-factor dimensions.

Stark conjecture. If $\varrho$ does not contain the trivial representation, then

$$L(1,\varrho)=\frac{W(\overline\varrho)2^a\pi^b}{f_\varrho^{1/2}}\,\theta(\overline\varrho)\det R(\overline\varrho),$$

for some $\theta(\overline\varrho)\in\mathbb{Q}(\operatorname{Tr}\varrho)^\times$.

This relates the special value at $s=1$ to logarithms of Stark units. In the paper it is used to interpret the rationality conjecture for cohomology classes.