  We generalize a theorem of D. Rohrlich concerning root numbers of elliptic
curves over the field of rational numbers. Our result applies to curves of all
higher genera over number fields. Namely, under certain conditions which
naturally extend the conditions used by D. Rohrlich, we show that the root
number associated to a smooth projective curve over a number field F and a
complex finite-dimensional irreducible representation of the absolute Galois
group of F with real-valued character is equal to 1. In the case where the
ground field is the field of rational numbers, we show that our result is
consistent with the refined version of the conjecture of Birch and
Swinnerton-Dyer.
