  The parity of the analytic rank of an elliptic curve is given by the root
number in the functional equation L(E,s). Fixing an elliptic curve over any
number field and considering the family of its quadratic twists, it is natural
to ask what the average analytic rank in this family is. A lower bound on this
number is given by the average root number. In this paper, we investigate the
root number in such families and derive an asymptotic formula for the
proportion of curves in the family that have even rank. Our results are then
used to support a conjecture about the average analytic rank in this family of
elliptic curves.
