Let $X$ be a smooth, closed, connected spin 4-manifold with $b_1(X)=0$ and non-positive signature $\sigma(X)$. In this paper we use Seiberg-Witten theory to prove that if $X$ admits a spin $Z_6$ action of even type, then $b^+_2(X)\geq|\sigma(X)|\backslash8+2$ under some non-degeneracy conditions.