Zeros near [formula omitted] and the constant term of [formula omitted] for [formula omitted]-functions in the Selberg class

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Let L(s)=∑n=1∞ann−s be an L-function in the Selberg class, and qL its conductor. Let ℓ0(L) be the constant term of the Laurent expansion of L′/L at s=1. We show that for certain families F of L-functions in the Selberg class with polynomial Euler product:• If L∈F has no zeros β+iγ with β>1−δ(logqL)−1, |γ|<(logqL)−1/2 for some absolute δ>0, then ℜ(ℓ0(L))≪FlogqL;• If ℜ(ℓ0(L))≪logqL for all L∈F, then there is some absolute δ>0 such that L has no zeros β+iγ with β>1−δ(logqL)−1, |γ|<(1−β)1/2(logqL)−1/2.This generalizes, for instance, the case of families of Dedekind zeta functions of number fields with bounded degree.