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.][1]

Kobayashi (Shin-ichi), *The local root number of elliptic curves with wild ramification*,
Math. Ann. 323 (2002), no. 3, 609–623. [Springer.][2]

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.][3]


  [1]: http://www.numdam.org/item?id=CM_1996__100_3_311_0
  [2]: https://doi.org/10.1007/s002080200318
  [3]: https://blms.oxfordjournals.org/content/40/3/516