nt.number theory - An everywhere locally trivial line bundle

The following example was provided to me by Colliot-Thélène some years ago : Let $X$ be the complement in $mathbb{P}_{1,mathbb{Q}}$ of the three closed points defined by $x^2=13$, $x^2=17$, $x^2=221$. Then $operatorname{Pic}(X)=mathbb{Z}/2mathbb{Z}$ but $operatorname{Pic}(X_v)=0$ for every place $v$ of $mathbb{Q}$.