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Income taxes
 6 Months Ended
Jun. 30, 2024
Income taxes [Abstract]  
Income taxes
8. Income taxes

We recorded an income tax benefit in the second quarter of 2024 of $86 thousand at an effective tax rate of (21.2%) compared to an income tax expense in the second quarter of 2023 of $391 thousand at an effective tax rate of 33.8%.  For the six months ended June 30, 2024, we recorded income tax benefit of $363 thousand at an effective tax rate of (21.1%), compared to an income tax expense for the six months ended June 30, 2023 of $1.0 million at an effective tax rate of 20.7%.
We are subject to U.S. federal income tax, as well as income tax in certain U.S. state and foreign jurisdictions.  We have substantially concluded all U.S. federal, state and local income tax, and foreign tax regulatory examination matters through 2019.  However, our federal tax returns for the years 2020 through 2023 remain open to examination. Various U.S. state and foreign tax jurisdiction tax years remain open to examination as well, but we believe that any additional assessment would be immaterial to the Condensed Consolidated Financial Statements.  The Company maintains a valuation allowance against certain deferred tax assets where realization is not certain.