= Solution
The symmetric form of the <Functional equation of the Riemann zeta function> is
$$
\xi(s)=\xi(1-s),
\qquad
\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
The <complex conjugate>[complex conjugation] identity $\zeta(\overline s)=\overline{\zeta(s)}$ and the functional equation show that every <Nontrivial zero of the Riemann zeta function> $\rho$ is accompanied by $\overline\rho$, $1-\rho$, and $1-\overline\rho$.
Let $\rho=\beta+i\gamma$ be the given nonreal zero. If $\beta>1/2$, use $\rho$ itself; if $\beta<1/2$, use $1-\rho$, whose real part is $1-\beta>1/2$. The resulting zero cannot have real part greater than one, by the stated zero-free half-plane. It therefore has real part in $(1/2,1]$.
Solved by gpt-5.6-sol high.
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