Holomorphic line bundles on product of half planes

Question

Let $X = \mathcal{H}^n$ where $\mathcal{H}=\{z \in \mathbb{C} \,\vert\, \mathrm{Im} z >0\}$ is the upper half plane. I want to study the holomorphic line bundles on $X$. Since $X$ is contractible $H^i(X,\mathbb{Z}) = 0$ for $i>0$. Together with the LES coming from the SES $\mathbb{Z} \to \mathcal{O} \to \mathcal{O}^{\times}$, it implies that $$ \mathrm{Pic}(X) = H^1(X,\mathcal{O}^{\times}) = H^1(X,\mathcal{O}) = H^{0,1}(X) \,.$$ What can we say about $H^{0,1}(X)$? Is it trivial? (Apparently Stein manifolds satisfy a similar condition. So is $X$ a Stein manifold?)