No. But maybe $9 or $19/month might be better.
You usually will learn a lot more from paid customers than free users, so when in doubt, charge.
But $1? I dunno about that idea.
Pricing is so confusing in SaaS. Why is Salesforce $200 a seat for Unlimited Edition — while Dropbox is $5? Are the bits better? The code higher quality? The server costs higher? No, no and no.
Pricing in SaaS is almost all about context. What do I also pay for other services of similar value.
And $1 is just too confusing. (Forget about the administrative and other costs).
Price your product within range of a comparable to learn if you have comparable value.
Price outside of context, and it’s hard to learn.