Let χ be a real and non-principal Dirichlet character, L(s, χ) its Dirichlet L-function and let p be a generic prime number. We prove the following result: If for some 0 ≤ σ < 1 the partial sums p≤x χ(p)p −σ change sign only for a finite number of x, then there exists > 0 such that L(s, χ) has no zeros in the half plane Re(s) > 1 − . Moreover, if f : N → [−1, 1] is a completely multiplicative function that is small on average, i.e., n≤x f (n) = o(x 1−δ ) for some δ > 0, and if the Dirichlet series of f , say F (s), is such that F (1) = 0, then there exists > 0 such that F (s) = 0 for all Re(s) > 1 − if and only if there exists 0 ≤ σ < 1 such that the partial sums p≤x f (p)p −σ change sign only for a finite number of x.