Amaj7b5 is A, C#, Eb and G# — with the fifth lowered a semitone.
It sounds hollow and hard to place. Far more often met spelled the other way round, as a ♯11 over a chord that kept its fifth.
Coloured tonic — rare, and unstable for a major chord.
Also written AM7b5, AΔ7b5 or Amaj7(b5).

How you get to it. The turn that lands on Amaj7b5 is to — the ii and the V of A — and shares its third and seventh with , so the hand barely moves between them.
Hear it land:→→
In which keys. Every note of Amaj7b5 belongs to C# minor (the ♭VI) and E major (the IV, the subdominant).
Every note of Amaj7b5 is in A lydian and A lydian augmented.
Each of those is a page: the scale across the neck, and the chords it harmonises into — which is where Amaj7b5 sits in it. Every scale.
Change one note of Amaj7b5 and you have (the flat fifth becomes the fifth), (the flat fifth becomes the sharp fifth), (the major third becomes the minor third), (the major seventh becomes the flat seventh) or (the flat fifth becomes the fifth).
Amaj7b5 is a major chord — root, major third, fifth. Major chords on every root.
The common shapes and the derived voicings come from different places, and both are explained in how this site is built.
Shows each chord the way this style would voice it, and marks which of its alternatives belong to the idiom.