Generally, financial institutions will accept $0 to fund its retirement accounts.
Note that if your dependent exemptions, deductions and other credits already reduce your taxable income to $0, an IRA contribution might not help you.
One of the benefits of having until 4/15 to make your IRA contribution is that you can "plug-in" different IRA amounts and see the effect on your refund before filing and before making the contribution. See at what point the IRA contribution no longer helps you and use that as the figure to take to the bank.
Or use the amount that helps you as a traditional IRA contribution and use the rest of the $3500 as a Roth IRA contribution. In a Roth IRA, you don't get to deduct your contributions. However, because Roth IRAs are funded with after-tax dollars (money that's already been taxed), you will pay taxes on your contributions but will not pay taxes when you withdraw them in the future, if you meet the age distribution requirements.
For traditional IRA contributions, your tax bracket would determine how much your contribution would affect your taxes. At a 15% tax bracket, a $1000 contribution would save you $150 in taxes, and so forth.
At lower income levels, you could qualify for the Saver's Credit for having a lower income and making a IRA contribution. You can claim the credit for 50%, 20% or 10% of the first $2,000 you contribute during the year to a retirement account. See this: https://ttlc.intuit.com/questions/3487688