UPPER-BOUNDS ON [L(1,CHI)] AND APPLICATIONS

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 plates. We also explain how we can compute better upper bounds and explain ho w useful such computed bounds are when dealing with class number probl ems for CM-fields. For example, we will reduce the determination of al l the non-abelian normal CM-fields of degree 24 with Galois group SL2( F-3) (the special linear group over the finite field with three elemen ts) which have class number one to the computation of the class number s of 23 such CM-fields.