Let K be a local field with finite residue class field and E a finite Galois extension over K. In this paper, we study the Artin conductor $f_Artin(X_{rho})$ of a character $X_{rho}$ associated to a representation $rho$:Gal(E/K) $rightarrow$ GL(V) of Gal(E/K) with metabelian kernel ker($rho$). In order to do so, we first review the Artin character $a_ {Gal} _ {(E/K)}$ of Gal(E/K) and review the metabelian local class field theory. We finally propose the definition of the conductor $f$(E/K) of a metabelian extension E/K in the sense of Koch-de Shalit local class field theory, and compute $f_{Artin}(X_{rho})$ under a suitable assumption.