I don't have a hard-fast rule for this. And instead of an "emergency fund" I think of the cash I keep on hand as an "operating account". Rent, late fees go in, regular cash outflows like insurance, mortgage, maintenance, utilities (when vacant) go out. But part of that account (or separate account) is the security deposit of each tenant. If you have 2 places that have rent/deposits of $1000 each, you can lose $2000 pretty quick if two tenants happen to move out at once. Don't forget to keep those liabilities readily available and maybe even in a dedicated account.
If you have (and plan to use) a HELOC for big ticket capital expenditures, you won't need a huge cushion.
I probably keep in the operating account, between 2.5 and 7.5% of the property value. For each $100K in property, $2500-7500 in addition to security deposit liabilities.