Euler characteristics of Tensor power of line bundle

Question

Let $X$ be a smooth projective variety, and $L$ a line bundle on $X$ which satisfies $L^{\otimes n}=O_X$ for some $n$. I am wondering whether there is a relation between $\chi(L)$ and $\chi(O_X)$. Something like $\chi(L)=\chi(O_X)^n$ seems too optimistic.

Answer 1

If $L^{\otimes n} \cong O_X$ then $L$ is numerically trivial, i.e., $c_1(L) = 0$ in $CH^1(X)_{num}$. Therefore, by Riemann--Roch $$ \chi(L) = \chi(O_X). $$