Uniqueness results of Dirichlet series with periodic coefficients

Let L(s,fj) with j∈{1,2} be a non-constant meromorphic function of finite order having a convergent Dirichlet series representation of the form L(s,fj)=∑n≥1fj(n)ns with j∈{1,2} in some right half-plane, where fj:Z→C is a q-periodic arithmetical function, and let a and b be two distinct finite complex values. If L(s,f1) and L(s,f2) share a CM and b IM, then L(s,f1)=L(s,f2) identically in C. Moreover, when L(s,fj),(j=1,2), admits an analytic continuation as a meromorphic function of finite order, we show some relations between L(s,f1), L(s,f2) if they share the set {a,b}.