Local root number (awaiting review)

The local root number of an elliptic curve $E$ over a number field at a prime $\mathfrak p$ is the sign of the local functional equation of $E$. This is equal to $+1$ at primes of good or non-split multiplicative reduction, $-1$ at primes of split multiplicative reduction, and at primes of additive reduction is $\pm1$ with no simple characterization.

For an elliptic curve defined over $\Q$ the root number at a prime $p$ is also equal to the eigenvalue of the associated Atkin-Lehner involution for the associated modular form.