Computing the average root number of an elliptic surface

By considering a one-parameter family of elliptic curves defined over $\mathbb{Q}$, we might ask ourselves if there is any bias in the distribution (or parity) of the root numbers at each specialization. From the work of Helfgott, we know (at least conjecturally) that the average root number of an elliptic curve defined over $\mathbb{Q}(T)$ is zero as soon as there is a place of multiplicative reduction over $\mathbb{Q}(T)$ other than $-$deg. Recently, Helfgott's work was extended by Desjardins, where she relaxes some of Helfgott's hypotheses and is able to provide unconditional results on the variation of the root number for many elliptic surfaces. In this paper, we are concerned with elliptic curves defined over $\mathbb{Q}(T)$ with no place of multiplicative reduction over $\mathbb{Q}(T)$, except possibly at $-$deg. More precisely, we will use the work of Helfgott to compute the average root number of an explicit family of elliptic curves defined over $\mathbb{Q}$ and show that this family is "parity-biased" infinitely-often.