Helson zeta functions for characters with finitely many values

We show that the analytic continuations of Helson zeta functions ζ χ ( s ) = ∑ 1 ∞ n − $ \zeta _\chi (s)= \sum _1^{\infty }\chi (n)n^{-s}$ can have essentially arbitrary poles and zeroes in strip 21 / 40 < ℜ 21/40 \Re 1$ (unconditionally), whole critical 2 1/2 <1$ under Riemann Hypothesis, with function \chi$ taking values cubic roots unity. If sets are symmetric respect to real axis, same be achieved ± \pm . The proof is constructive.