Let us consider the Fubini-Study metric on the part at infinity of the line bundle $\mathcal{O}_{\mathbb{P}^1(\mathbb{Z})}(1)$ to obtain the Hermitian line bundle $\overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})}(1)$.
Now we can consider the arithmetic self intersection number $\overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})}(1).\overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})}(1) $, and in this article it is implicitly stated the value $$ \overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})} (1). \overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})} (1)= -\frac{1}{2}.$$
On the other hand, if I try to explicitly compute it I get, choosing for example the sections associated to the homogeneus polynomials $X_0$ and $X_1$: $$ \overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})}(1) . \overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})} (1) = 0 - \int_{\mathbb{P}^1(\mathbb{C})} \log(\|X_0\|_{FS})\,c_1(\overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{C})}(1))=1.$$
The computation should be quite a standard example, but I didn't manage to find any other indication on the web. What is the correct value?
Thanks!