Equivalently, under the standard identification $J(X)(\mathbb{C})=\operatorname{Pic}^0(X)$, a divisor class represented by a degree-zero divisor $D=\sum_{i=1}^r n_i[P_i]$ is sent to the class of the functional
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\begin{align*}
V &\longrightarrow \mathbb{C} \\
\omega &\longmapsto \sum_{i=1}^r n_i\int_{P_0}^{P_i}\omega
\end{align*}
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modulo the period lattice, where $P_0\in X$ is any fixed base point and the path integrals are taken along arbitrary piecewise smooth paths from $P_0$ to $P_i$.