Helson zeta functions for characters with finitely many values

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 for the function $\chi$ taking values in cubic roots of unity. If the sets are symmetric with respect to the real axis, the same can be achieved with $\chi$ taking values $\pm 1$.