Given a digraph $ G $, a weight was assigned to each vertex. For this digraph, a vertex weighted zeta function was defined, the weight of which was associated with the weights of the cycles in $ G $, induced by the vertex weights given above. Two determinant expressions of the vertex weighted zeta function were provided, using Amitsur identity and some techniques in linear algebra. When the digraph was symmetric, the existing results were natural corollaries of our results. Finally, an example was given and the specific form of the vertex weighted zeta function was calculated.