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Let $\chi(s)=2^s \pi^{s-1}\sin(\pi s/2) \Gamma(s)$ Gamma(1-s)$ so that $\zeta(s)=\chi(s)\zeta(1-s)$. You are asking about the curve $|\chi(s)|=1$.

As you have observed, $|\chi(1/2+it)|=1$ for real $t$. There is a partial converse to this statement, namely that there is a positive absolute constant $C_0$ such that if $|\chi(\sigma+it)| = 1$ with $0 \le \sigma \le 1$ and $|t| \ge C_0$, then $\sigma=1/2$.

A simple proof can be found in Lemma 6.1 of S. M. Gonek "Finite Euler products and the Riemann hypothesis" Trans. Amer. Math. Soc. 364 (2012), 2157-2191. This paper is also on the arXiv. Gonek states that $C_0<6.3$ so it seems that phenomena in your pictures stops shortly after the ranges you plotted.
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Let $\chi(s)=2^s \pi^{s-1}\sin(\pi s/2) \Gamma(s)$ so that $\zeta(s)=\chi(s)\zeta(1-s)$. You are asking about the curve $|\chi(s)|=1$.

As you have observed, $|\chi(1/2+it)|=1$ for real $t$. There is a partial converse to this statement, namely that there is a positive absolute constant $C_0$ such that if $|\chi(\sigma+it)| = 1$ with $0 \le \sigma \le 1$ and $|t| \ge C_0$, then $\sigma=1/2$.

A simple proof can be found in Lemma 6.1 of S. M. Gonek "Finite Euler products and the Riemann hypothesis" Trans. Amer. Math. Soc. 364 (2012), 2157-2191. This paper is also on the arXiv. Gonek states that $C_0<6.3$ so it seems that phenomena in your pictures stops shortly after the ranges you plotted.