In the specific context of elliptic curves $C$ over (any finite extension of) $\mathbf{Q}$, the root number has been analysed by
Rohrlich (David), Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), no. 3, 311–349. Numdam.
Kobayashi (Shin-ichi), The local root number of elliptic curves with wild ramification, Math. Ann. 323 (2002), no. 3, 609–623. Springer.
and
Dokchitser (Tim) & Dokchitser (Vladimir), Root numbers of elliptic curves in residue characteristic 2, Bull. Lond. Math. Soc. 40 (2008), no. 3, 516–524. Oxford Journals.
