On zeroes and poles of Helson zeta functions

We show that the analytic continuations of Helson zeta functions $\zeta_\chi (s)= \sum_1^{\infty}\chi(n)n^{-s}$ can have essentially arbitrary poles and zeroes in the strip $21/40 < \Re s < 1$ (unconditionally), and in the whole critical strip $1/2 < \Re s <1$ under Riemann Hypothesis.