Integral first Chern class of the line bundle associated with a character

Question

Let $X$ be a connected complex projective manifold, $\chi:\pi_1(X)\to S^1$ be a character of the fundamental group of $X$. Then $\chi$ induces a local system $\mathcal{L}_{\chi}$ of rank $1$ on $X$ and $L=\mathcal{L}_{\chi}\otimes_{\mathbb{C}}O_X$ is a line bundle on $X$. Since $L$ is flat, the real Chern class $c_1^{\mathbb{R}}(L)=0$ in $H^2(X,\mathbb{R})$. What is the integral Chern class $c_1^{\mathbb{Z}}(L)$ in $H^2(X,\mathbb{Z})$? It must be an torsion element, but is there an explicit formula?