The question is in the title: can a Landau-Siegel zero be the only zero off the critical line for a Dirichlet L-function or does its existence imply the existence of a complex non trivial zero in the critical strip off the critical line?


This question came to my mind considering the sequence of trivial zeros in decreasing order for zeta as a $2$-periodic signal whose Fourier transform would be a $1/2$-periodic signal made of Dirac peaks (the former future physicist in me is speaking, sorry), which if we compactify partially the critical strip by identifying the vertical lines of real parts $0$ and $1$ (this compactification process should be compatible with the functional equation as it relates the value at $s$ to the one at $1-s$) becomes a single Dirac peak which is supported on $1/2$, hence the real part of the non trivial zeros under RH. So my idea is that adding a Landau-Siegel zero would create a non periodic signal made by the decreasing sequence of real zeros, whose Fourier transform would not be periodic either, suggesting the existence of a complex non trivial zero in the critical strip off the critical line. In some sense, the Fourier transform is expected to *permute* zeros of L-functions, which seems consistent with the Deuring-Heilbronn phenomenon.


So would the existence of a Landau-Siegel zero create such a havoc that the analogue of RH for the considered Dirichlet L-function would fail completely?



Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the critical line?
A result of Sarnak and Zaharescu, stated
in the contrapositive, implies the existence of a complex zero off the critical line for at least one Dirichlet L-function if one has a sufficiently strong Siegel zero: [ProjectEuclid link](https://www.projecteuclid.org/journals/duke-mathematical-journal/volume-111/issue-3/Some-remarks-on-Landau-Siegel-zeros/10.1215/S0012-7094-02-11134-X.short)


