The zeros of Dedekind zeta functions and class numbers of CM-fields

Let $F'/F$ be a finite normal extension of number fields with Galois group $Gal(F'/F)$. Let $\chi$ be an irreducible character of $Gal(F'/F)$ of degree greater than one and $L(s,\chi)$ the associated Artin $L$-function. Assuming the truth of Artin's conjecture, we have explicitly determined a zero-free region about $1$ for $L(s,\chi)$. As an application we show that, for a CM-field $K$ of degree $2n$ with solvable normal closure over $\mathbb{Q}$, if $n \geq 370$ as well as $n \notin \{ 384, 400, 416, 448, 512 \}$, then the relative class number of $K$ is greater than one.