Some remarks on the unique factorization in certain semigroups of classical $L$-functions

Abstract

In this note we investigate problems related to the unique factorization of some semigroups of classical $L$-functions. The semigroups of Artin and automorphic $L$-functions as well as the semigroup generated by the Hecke $L$-functions of finite order are studied. The main result of the paper shows that in the latter semigroup the unique factorization into primitive elements does not hold. This closes a possible way of attacking the famous Dedekind conjecture concerning the divisibility of the Dedekind zeta functions.