On the nature of eγ and non-vanishing of derivatives of L-series at s=1/2
In 2011, M.R. Murty and V.K. Murty [10] proved that if L(s, χD) is the Dirichlet L-series attached a quadratic character χD, and L'(1, χD)=0, then eγ is transcendental. This paper investigates such phenomena in wider collections of L-functions, with a special emphasis on Artin L-functions. Instead of s=1, we consider s=1/2. More precisely, we prove thatexp (L'(1/2,χ)L(1/2,χ)-αγ) is transcendental with some rational number α. In particular, if we have L(1/2, χ)≠0 and L'(1/2, χ)=0 for some Artin L-series, we deduce the transcendence of eγ.
