Euler characteristic of odd-dimensional compact connected orientable manifold is zero

Statement
Suppose $$M$$ is a fact about::compact connected orientable manifold that has odd dimension. Then, $$M$$ is a fact about::space with zero Euler characteristic, i.e., the fact about::Euler characteristic of $$M$$ is zero.

Related facts

 * Compact connected nontrivial Lie group implies zero Euler characteristic