Haselgrove condition for the modulus $k$

Let $k$ be a modulus, let $\chi$ range over Dirichlet characters modulo $k$, and write $s=\sigma+it$. Haselgrove condition. There is a function $Z(k)$ with $0<Z(k)\leq 1$ such that no $L(s,\chi)$ with $\chi$ modulo $k$ vanishes for

$$0<\sigma<1,\qquad |t|\leq Z(k).$$

The condition gives a zero-free region near the real segment of the critical strip and is used in comparisons of primes in residue classes. The source presents it as a property known for the listed moduli, rather than establishing a general conjectural status.