Let $$extract_itex$$(M,psi)$$/extract_itex$$ be a $$extract_itex$$(2n+1)$$/extract_itex$$-dimensional oriented closed manifold equipped with a pseudo-free $$extract_itex$$S^1$$/extract_itex$$-action $$extract_itex$$psi : S^1 times M rightarrow M$$/extract_itex$$. We first define a textit{local data} $$extract_itex$$mathcal{L}(M,psi)$$/extract_itex$$ of the action $$extract_itex$$psi$$/extract_itex$$ which consists of pairs $$extract_itex$$(C, (p(C) ; overrightarrow{q}(C)))$$/extract_itex$$ where $$extract_itex$$C$$/extract_itex$$ is an exceptional orbit, $$extract_itex$$p(C)$$/extract_itex$$ is the order of isotropy subgroup of $$extract_itex$$C$$/extract_itex$$, and $$extract_itex$$overrightarrow{q}(C) in (Z_{p(C)}^{times})^n$$/extract_itex$$ is a vector whose entries are the weights of the slice representation of $$extract_itex$$C$$/extract_itex$$. In this paper, we give an explicit formula of the Chern number $$extract_itex$$langle c_1(E)^n, $$M/S^1$$ rangle$$/extract_itex$$ modulo $$extract_itex$$Z$$/extract_itex$$ in terms of the local data, where $$extract_itex$$E = M times_{S^1} C$$/extract_itex$$ is the associated complex line orbibundle over $$extract_itex$$M/S^1$$/extract_itex$$.Also, we illustrate several applications to various problems arising in equivariant symplectic topology.