  This paper studies zeta functions of the form $\sum_{n=1}^{\infty} \chi(n)
n^{-s}$, with $\chi$ a completely multiplicative function taking only
unimodular values. We denote by $\sigma(\chi)$ the infimum of those $\alpha$
such that the Dirichlet series $\sum_{n=1}^{\infty} \chi(n) n^{-s}$ can be
continued meromorphically to the half-plane $\operatorname{Re} s>\alpha$, and
denote by $\zeta_{\chi}(s)$ the corresponding meromorphic function in
$\operatorname{Re} s>\sigma(\chi)$. We construct $\zeta_{\chi}(s)$ that have
$\sigma(\chi)\le 1/2$ and are universal for zero-free analytic functions on the
half-critical strip $1/2<\operatorname{Re} s <1$, with zeros and poles at any
discrete multisets lying in a strip to the right of $\operatorname{Re} s =1/2$
and satisfying a density condition that is somewhat stricter than the density
hypothesis for the zeros of the Riemann zeta function. On a conceivable version
of Cram\'{e}r's conjecture for gaps between primes, the density condition can
be relaxed, and zeros and poles can also be placed at $\beta+i \gamma$ with
$\beta\le 1-\lambda \log\log |\gamma|/\log |\gamma|$ when $\lambda>1$. Finally,
we show that there exists $\zeta_{\chi}(s)$ with $\sigma(\chi) \le 1/2$ and
zeros at any discrete multiset in the strip $1/2<\operatorname{Re} s \le 39/40$
with no accumulation point in $\operatorname{Re} s >1/2$; on the Riemann
hypothesis, this strip may be replaced by the half-critical strip $1/2 <
\operatorname{Re} s < 1$.

Number Theory;Complex Variables