Locally ringed point space is not a scheme?

Question

Let $X$ be the point space $\{*\}$, and let the structure sheaf $\mathcal{O}_X$ be given by: $$\mathcal{O}_X (X) = \mathcal{O}_X (\{*\}) := \mathbb{Z}_p$$ $$\mathcal{O}_X (\varnothing) := 0$$ Why is it the case that $(X, \mathcal{O}_X)$ is only a locally ringed space and not a scheme ?