Wow I just got my site crawled yesterday afternoon and I'm #3 for my search word, not too shabby. Just goes to show how important the domain name is in SERPs
Domain name is a ranking factor but it's not the only one nor that important compared to backlinks. However, getting ranking high quickly is one factor but staying there is the hard job.. Based on the niche you're targeting expect your website to fall down the ranking
I guess the competition is really low for that keyword then. If its a new site chances are very high it will go down soon. You better start working on building links.
Speaking from my experience, domain name is a VERY important factor when it comes to initial ranking (yes, I say initial because without proper backlinking the ranking usually does seem to drop after a month or two). A very good example was a site that I almost launched - by 'almost' I mean that there was a Wordpress installation on this site, containing the name of the site and the 'Hello World' post and that's it. After checking the SERPs around 1 month after I had launched the site, it was 1st for my keyword. Obviously it wasn't a very competitive keyword but not the easiest one either. Then, after building the site up and adding a content (but getting comfortable with my SERPs and therefore doing no backlinking), I suddenly discovered one day that my site was on the 3rd page. Lesson learned
proper naming for the URL of the site is a factor and is in the process of on-page optimization. its important to place a keyword/s on it, as also others says.