Formulae for some Artin root numbers

Let N be a normal number field with Galois group any dicyclic group of order 4p, p greater than or equal to 3 an odd prime. Let L denote the real quadratic subfield of N. We determine the root numbers which appear in the functional equations of the Artin L-functions L(s, psi, N/Q) associated with the irreducible characters of degree two of the Galois group Gal(N/Q). We deduce that if p is totally ramified in N and L = Q(rootp), then the Dedekind zeta function of N has a zero of order greater than or equal to (p - 1)/2 at s = 1/2.