The C-7b5 chord (also known as C half diminished 7 and CØ7).
The notes in C-7b5 are:
An arpeggio is a succession of individual notes that spell the chord. Each note spells the chord and is called a chord tone.
The Half Diminished Scale
The half diminished scale is perfect to use over C-7b5
C-7b5 Backing Track
Practice playing the half diminished scale while listening to C-7b5. See what lines you can play over it. To train your ear, make a sincere effort to internalise the sound.
If you’re having trouble coming up with new ideas, my books are filled with several ideas on how to play intervals, scales, melodies, and more.
1st Inversion - C-7b5/Eb
2nd Inversion - C-7b5/Gb
3rd Inversion - C-7b5/Bb
4 Way Close
We simply move the 2 voice an octave down.
For Drop 3, we simply move the 3rd voice down an octave.
Drop 2 + 3
For drop 2+3, we move the 2nd + 3rd voices down an octave.
Drop 2 + 4
For Drop 2+4, we move the 2nd + 4th voice down an octave.
These were the drop voicings for the normal voicing; however, the first, second, and third inversions can also be turned into drop voicings. If you want to study more about drop voicings, Rick Beato has a solid tutorial on it.