Special values of Abelian L-functions at s=0

In [H.M. Stark, L-functions at s=1. IV. First derivatives at s=0, Adv. Math. 35 (3) (1980) 197–235], Stark formulated his far-reaching refined conjecture on the first derivative of abelian (imprimitive) L-functions of order of vanishing r=1 at s=0. In [Karl Rubin, A Stark conjecture “over Z” for abelian L-functions with multiple zeros, Ann. Inst. Fourier (Grenoble) 46 (1) (1996) 33–62], Rubin extended Stark's refined conjecture to describe the rth derivative of abelian (imprimitive) L-functions of order of vanishing r at s=0, for arbitrary values r. However, in both Stark's and Rubin's setups, the order of vanishing is imposed upon the imprimitive L-functions in question somewhat artificially, by requiring that the Euler factors corresponding to r distinct completely split primes have been removed from the Euler product expressions of these L-functions. In this paper, we formulate and provide evidence in support of a conjecture in the spirit of and extending the Rubin–Stark conjectures to the most general (abelian) setting: arbitrary order of vanishing abelian imprimitive L-functions, regardless of their type of imprimitivity. The second author's conversations with Harold Stark and David Dummit (especially regarding the order of vanishing 1 setting) were instrumental in formulating this generalization.