Helson zeta functions for characters with finitely many values

We show that the analytic continuations of Helson zeta functions ${\zeta }_{\chi }(s)={\sum }_1^{\mathrm{\infty }}\chi (n)n^{-s}$ can have essentially arbitrary poles and zeroes in the strip $21\mathrm{/}40<\mathrm{\Re }s<1$ (unconditionally), and in the whole critical strip $1\mathrm{/}2<\mathrm{\Re }s<1$ 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 $\pm 1$.