Let $M^4$ be an orientable closed 4-manifold and $c_1$ be the first Chern class of a complex line bundle on $M^4$. Let $b$ be the mod 2 reduction of $c_1$, ie $b=c_1$ mod 2.

We have a relation $w_2 b = b^2$, where $w_n$ is the $n^\text{th}$ Stiefel-Whitney class of the tangent bundle of $M^4$. This implies that if $M^4$ is spin, then the Chern number on $M^4$ must be even, ie $\int_{M^4} c_1^2 =0$ mod 2.

My question is that for any $M^4$ that is not spin, can we always find a complex line bundle on $M^4$, such that the Chern number on $M^4$ is odd, ie $\int_{M^4} c_1^2 =1$ mod 2.