Zener diodes aren't switches so don't turn on.
Some zener diodes are designed to be used at 45mA. At 5mA they regulate poorly with a voltage much less than their rating and are almost a resistor.
Other zener diodes are designed for 5mA and up. They reguate well at nearly any current.
I have a book printed in 1968 with datasheets showing voltage vs current. New datasheets don't have those curves so just measure a zener diode yourself.
Here is a comparison of zener diodes:
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