A new Bartholdi zeta function of a digraph

We introduce a new type of the Bartholdi zeta function of a digraph D, and give a determinant expression of it. Furthermore, we define a new type of the Bartholdi L-function of D, and give a determinant expression of it. As a corollary, we obtain a decomposition formula for a new type of the Bartholdi zeta function of a group covering of D by new Bartholdi L-functions of D.