The fibrewise trivial locus of a line bundle

Let $\mathcal{L}$ be a line bundle on a smooth and proper scheme $X$ over $S$. We compute, in the case where $S$ is smooth over a field of characteristic $0$, the virtual fundamental class of the closed subset of $S$ consisting of those $s\in S$ such that the restriction of $\mathcal{L}$ to the fibre $X_s$ over $s$ is trivial.