Take the following definite integral:

$$f(s):=\int_s^{1-s} \zeta(x) \mathrm{d} x$$

with $s \in \mathbb{C}$, $s=\sigma \pm ti$, $0<\sigma<1$ and $t,\sigma \in \mathbb{R}$.

The graph of $|f(s)|$ shows a monotonically increasing function for $\sigma=\frac12$ (as expected, it 'plateaus' exactly at the $\rho$s) and an apparently strictly increasing function when $\sigma\ne\frac12$.

There is however a small 'dip' in the area $1 < t < 3$, that unexpectedly induces a zero at $\frac12 \pm 2.50056818181399528638615277529..i$. For $\sigma\ne\frac12$ there are no zeros.

Is there anything known about this zero? Could it be proven that it only exists for $\sigma=\frac12$?

Thanks!