Effective Brauer-Siegel theorems for Artin $L$-functions

Given a number field $K \neq \mathbb{Q}$, in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function $\zeta_K(s)$ and used this to give effective upper and lower bounds on the residue of $\zeta_K(s)$ at $s=1$.

We extend Stark's work to give effective upper and lower bounds for the leading term of the Laurent expansion of general Artin $L$-functions at $s=1$ that are, up to the value of implied constants, as strong as could reasonably be expected given current progress toward the generalized Riemann hypothesis.

Our bounds are completely unconditional, and rely on no unproven hypotheses about Artin $L$-functions.