We give upper bounds on the modulus of the values at $s=1$ of Artin $L$-functions of abelian extensions unramified at all the infinite places. We also explain how we can compute better upper bounds and explain how useful such computed bounds are when dealing with class number problems for $\CM$-fields. For example, we will reduce the determination of all the non-abelian normal $\CM$-fields of degree $24$ with Galois group $\SL_2(F_3)$ (the special linear group over the finite field with three elements) which have class number one to the computation of the class numbers of $23$ such $\CM$-fields.

Keywords:

Dedekind zeta function, Dirichlet series, $\CM$-field, relative class number Dedekind zeta function, Dirichlet series, $\CM$-field, relative class number

MSC Classifications:

11M20, 11R42, 11Y35, 11R29 show english descriptions Real zeros of $L(s, \chi)$; results on $L(1, \chi)$
Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]
Analytic computations
Class numbers, class groups, discriminants 11M20 - Real zeros of $L(s, \chi)$; results on $L(1, \chi)$
11R42 - Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]
11Y35 - Analytic computations
11R29 - Class numbers, class groups, discriminants

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