For symplectic type characters of wildly ramified quadratic extensions of p-adic fields, the epsilon factor sign is χ(-1)^d when the conductor is even and χ(-1)^l times a conjecturally ±1 Gauss sum when the conductor is odd.
Prasad, On an extension of a theorem of Tunnell, Compositio Math
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Epsilon factors of symplectic type characters in the wild case
For symplectic type characters of wildly ramified quadratic extensions of p-adic fields, the epsilon factor sign is χ(-1)^d when the conductor is even and χ(-1)^l times a conjecturally ±1 Gauss sum when the conductor is odd.