Let $X$ be a normal variety over a field $k$, $L$ is a line bundle on it, and $s$ is a global section of $L$. If the Weil divisor associated to $s$ equals zero. Then can I conclude that $s$ is nonvanishing everwhere?

Equivalently, let $A$ be a integrally closed finitely generated $k$-algebra, and $s\in A$, if $s$ is not a unit, then can I conclude that there exits a prime idea $p$, such that

- dim($A_p$)=1
- the valuation of $s$ corresponding to $p$ is non-zero.

If it is not true in general, what extra conditions should I impose on the variety $X$?