The main purpose of this article is to present a generalization Forelli’s theorem for the functions holomorphic along general pencil discs. This generalizes result Joo et al. (Math Ann 355(3):1171–1176, 2013) and original theorem, furthermore, answers one problems posed in Chirka (Proc Steklov Inst Math 253(2):212–220, 2006).

Let K/Q be a finite Galois extension, s0 ∈ C \ {1}, Hol(s0) the semigroup of Artin L-functions holomorphic at s0. We present criteria for Artin’s holomorphy conjecture in terms of the semigroup Hol(s0). We conjecture that Artin’s L-functions are holomorphic at s0 if and only if Hol(s0) is factorial. We prove this if s0 is a zero of an L-function associated to a linear character of the Galois gr...