is Euler characteristic with coefficients in local system equal usual Euler characteristic times rank, for non-compact spaces?

Question

[This question is prompted by Proposition 2.5.4 (ii) in Dimca, "Sheaves in topology"]

Let $L$ be a local system on a topological space $X$ and assume that $L$ trivializes after pullback to a finite covering of $X$. Is it true then that $$ \chi(X,L) = \mathrm{rank}\ L \cdot \chi(X, \mathbb{Z}) $$

(Note that I do not assume that $X$ is a finite CW complex)