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Let $k$ be a number field and $G$ be a finite group. Let$\mathfrak{F}_{k}^{G}(Q)$ be the family of number fields $K$ with absolutediscriminant $D_K$ at most $Q$ such that $K/k$ is normal with Galois groupisomorphic to $G$. If $G$ is the symmetric group $S_n$ or any transitive groupof prime degree, then we unconditionally prove that for all$K\in\mathfrak{F}_k^G(Q)$ with at most $O_{\epsilon}(Q^{\epsilon})$ exceptions,the $L$-functions associated to the faithful Artin representations of$\mathrm{Gal}(K/k)$ have a region of holomorphy and non-vanishing commensuratewith predictions by the Artin conjecture and the generalized Riemannhypothesis. This result is a special case of a more general theorem. Asapplications, we prove that:  1) there exist infinitely many degree $n$ $S_n$-fields over $\mathbb{Q}$whose class group is as large as the Artin conjecture and GRH imply, settling aquestion of Duke;  2) for a prime $p$, the periodic torus orbits attached to the ideal classesof almost all totally real degree $p$ fields $F$ over $\mathbb{Q}$equidistribute on$\mathrm{PGL}_p(\mathbb{Z})\backslash\mathrm{PGL}_p(\mathbb{R})$ with respectto Haar measure;  3) for each $\ell\geq 2$, the $\ell$-torsion subgroups of the ideal classgroups of almost all degree $p$ fields over $k$ (resp. almost all degree $n$$S_n$-fields over $k$) are as small as GRH implies; and  4) an effective variant of the Chebotarev density theorem holds for almostall fields in such families.

An approximate form of Artin's holomorphy conjecture and non-vanishing of Artin $L$-functions