Zeros and poles of the Helson zeta function

Let $\chi\colon \mathbb N\to \mathbb T$, $\mathbb T = \{z\in\mathbb C\colon|z|=1\}$, be a multiplicative function. The Helson zeta function $\zeta_\chi$ is defined as:
\begin{equation}\label{H} \zeta_\chi(s)=\sum_1^{\infty}\chi(n)n^{-s} . \end{equation}
. The Riemann zeta function is thus a special case of the Helson zeta function.
With this definition, $\zeta_\chi$ is an analytic function in the half-plane $\operatorname{Re} s > 1$, for which one can also write the Euler product
$$ \zeta_\chi(s)=\prod_p\frac{1}{1-\chi(p)p^{-s}} . $$
In particular, $\zeta_\chi$ has no zeros in the region $\operatorname{Re} s > 1$.
Result of X. Helson (X . H. Helson, Compact groups and Dirichlet series, Ark. Mat. 8 (1969), 139–143.) asserts that for almost all $\chi$'s the function $\zeta_\chi$ continues analytically over the domain $\operatorname{Re} s > 1/2$ and the continuation has no zeros in this domain. Nevertheless, as the example of the Riemann zeta function shows, the Helson zeta function can extend to large areas, and also have zeros or poles in them. This report will tell you that the set of zeros and poles of the Helson zeta functions in the band $1> \operatorname{Re} s > 1/2$ can be almost anything.
Namely, for any potential sets of zeros and poles (without accumulation points, which is a necessary condition for an analytical function) in the band $1>\operatorname{Re} s > 21/40$, there is a Helson zeta function with these sets of zeros and poles. Moreover, the function $\chi$ can be considered to accept only three values. Assuming the validity of the Riemann hypothesis, $21/40$ can be replaced by $1/2$.