For 1≤i≤r, let χi be primitive Dirichlet characters modulo qi and Z(t,χi) be the Z-function corresponding to the Dirichlet L-series L(s,χi). Let Ω(t) be a real linear combination of Z(t,χi). Since Z(t,χi) is real for real t, Ω(t) is real for real t. In this paper, we show that the Lebesgue measure of the set, where the functional values of Ω(t) is positive or negative in the interval [T,2T] is at least Tr2. We also study the Lebesgue measure of the set that the certain complex linear combinations of Z(t,χi) takes positive or negative values respectively. In particular, we study the distribution of signs of the Z-function correspond to the Davenport-Heilbronn function. Moreover, we prove that for sufficiently large T, the generalized Davenport-Heilbronn function has at least H(logT)2φ(q)−ϵ odd order zeros along the critical line on the interval [T,T+H].