Do I owe state tax?

[TunyaI]

Here are the details about state income taxes - these vary depending on what state you lived in throughout the year

https://turbotax.intuit.com/tax-tips/state-taxes/what-are-state-income-taxes/L4O5May37 

[MarkZ]

If the state where you reside has a personal income tax then you should have had state income taxes withheld from your wages reported on the W-2 in box 17.