The Lifted Root Number Conjecture (LRNC) is a famous conjecture which relates the leading terms at zero of Artin L-functions attached to a finite Galois extension L/K of number fields to natural arithmetic invariants. It has been introduced by K.W. Gruenberg, J. Ritter and A. Weiss in 1999. The conjecture depends on a set S of primes of L which is supposed to be sufficiently large. We formulate a LRNC for small sets S which only need to contain the archimedean primes and apply this to CM-extensions which we require to be (almost) tame above a fixed odd prime p. In this case the conjecture naturally decomposes into a plus and a minus part, and it is the minus part for which we prove the LRNC at p for an infinite class of relatively abelian extensions. Moreover, we show that the minus part of the LRNC implies the Strong Brumer-Stark Conjecture in the tamely ramified case.