Relation of Euler number of a line bundle over a Riemann surface and a section of it with finitely many isolated singularities

Question

Suppose $M$ is a compact Riemann surface and $E\to M$ a holomorphic line bundle over $M$. Also suppose $s:M\to E$ is a section with finitely many isolated singularities (undefined points, and not removable). At each of these singularities, we can define the order of $s$ since we can locally regard $s$ as a holomorphic complex function. Is there a relation of these orders and the euler number of $E$? (I'm expecting the following result: $e(E)$ is the sum of these orders, since there is a similar result relating the euler number of a vector bundle and a section with finitely many zeros.)