Are vertical minimum-modulus branches of the Riemann xi function a studied object?

I have been doing a computational/visual exploration of the Riemann zeta function and its completed xi function. I am a software developer and mathematics enthusiast rather than a professional mathematician, and my main aim here is to identify the established theory behind the following construction.

I am not claiming a proof of the Riemann Hypothesis.

Define

$$ F(\sigma,t)=|\xi(\sigma+it)|^2, $$

where

$$ \xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s). $$

For fixed $\sigma$, consider local minima of $F(\sigma,t)$ with respect to $t$. A smooth stationary branch $t=t_{\min}(\sigma)$ therefore satisfies

$$ \frac{\partial F}{\partial t}=0, $$

equivalently

$$ \operatorname{Im}\left(\xi'(s)\overline{\xi(s)}\right)=0. $$

Numerically continuing such a local-minimum branch through a known simple zero

$$ \rho=\frac12+i\gamma $$

produces a regular minimum-modulus valley. If

$$ V(\sigma)=\xi\left(\sigma+i\,t_{\min}(\sigma)\right), $$

the two sides of this branch are mirror images in the xi-plane.

Xi first zero valley here

Example: a numerically continued vertical minimum branch through the first nontrivial zero, mapped into the xi-plane. This figure motivated the question; I am not presenting its symmetry as numerical evidence for RH.

The mirror symmetry itself is explained by the functional equation. Writing

$$ \sigma=\frac12+\delta, $$

the symmetries of $\xi$ give

$$ \xi\left(\frac12-\delta+it\right) = \overline{\xi\left(\frac12+\delta+it\right)}. $$

Hence, for corresponding minimum branches,

$$ t_{\min}\left(\frac12-\delta\right) = t_{\min}\left(\frac12+\delta\right) $$

and

$$ V(-\delta)=\overline{V(\delta)}. $$

The local V-shaped minimum is also readily explained. Expanding about a simple zero,

$$ \xi(\rho+w) = \xi'(\rho)w+\frac12\xi''(\rho)w^2+\cdots, $$

gives to first order

$$ \min_t|\xi(\sigma+it)| \sim |\xi'(\rho)|\left|\sigma-\frac12\right|. $$

So I do not regard either the V shape or its symmetry as evidence for RH by themselves.

What I have not been able to determine is whether the global geometry of these vertical stationary/minimum branches is an established object.

What I have looked for

I have checked the standard material on the functional equation and xi function in the NIST DLMF, and looked at literature concerning level curves of the zeta function.

In particular, I found the G-curve/Z-curve terminology for level curves associated with Gram points and zeros, including Arias-de-Reyna's X-ray of Riemann's zeta function, and more recent work applying analogous level-curve ideas to functions related to $\zeta'(s)$.

These appear related geometrically, but I have not found a treatment specifically of components of

$$ \operatorname{Im}\left(\xi'(s)\overline{\xi(s)}\right)=0 $$

selected by the additional condition that they are local minima in the vertical direction.

My questions are therefore:

1. Is there an established name or theory for these vertical stationary/minimum-modulus branches of $|\xi|$?

2. Are there known results describing their continuation, bifurcations or topology in the critical strip?

3. In particular, is anything known about the conditions under which such a local-minimum branch can contain a zero

$$ \beta+i\gamma,\qquad \beta\ne\frac12? $$

I realise that at any zero ($\xi(s)=0$), the stationary equation above holds automatically, so that equation alone clearly cannot exclude an off-critical-line zero. My question is whether requiring continuation as a vertical local-minimum branch, rather than merely satisfying the stationary equation at one point, connects to any known useful theory.

For references/context I have so far used:

• NIST DLMF, §25.4, for the definition and functional equation of $\xi$: https://dlmf.nist.gov/25.4
• J. Arias-de-Reyna, X-ray of Riemann's zeta function, arXiv:math/0309433.
• Recent level-curve work using analogous G/Z-curve terminology: Level Curves for Zhang's Eta Function, Experimental Mathematics (2025).

I would particularly welcome references or terminology that would help me determine whether I am simply rediscovering an established object.

For provenance: I used ChatGPT extensively as a computational/research assistant during this exploration, including numerical experiments, checking local expansions, attempting to falsify apparent patterns, and locating relevant existing mathematics. I mention this explicitly rather than presenting the investigation as unaided.