I've seen this fact mentioned often in the literature, but I've never seen a proof.
Can someone suggest a reference where this is proved?
I haven't found it in the usual places (Iwaniec-Kowalski, Davenport...)
2026-04-06 03:38:27.1775446707
Siegel zeros are simple and real
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On pages 91-92, Davenport considers the case of a complex zero β+iγ (and then also its conjugate.) The same reasoning can be applied to two real zeros (including a double zero), showing that if there is a Siegel zero, then it is necessarily real and simple. This fact is also contained in Theorem 11.3 (beginning on page 360) of Montgomery and Vaughan.