that's probably because google shows you what THEY want you to see (google shows different results to different people, based on those people's search history, their thinking and behavioral patterns, google's agenda, etc)...
just use a VPN or proxy to check, it's safer....
I mean, if your domain contains words that are prone to trademark infringement or DMCA takedowns, it's possible that someone who owned this domain before you has been penalized by google with trademark / DMCA takedowns and now google refuses to show this domain name in their organic searches.
For example, if your domain name contains words like apple, microsoft, sony, disney, etc... These words belong (weird, I know, but that's the world we live in) to companies like Apple, Disney, Sony, etc.... And these companies don't want you to make money off their brand so they tell google "hey google, look at this good looking chap trying to make money off our brand! Can you, pretty please, get rid of his website so that people only buy from us instead of him because we are even more good looking than him and - more importantly - we have truck loads of money that we can share with you? So, how about that, google? Sincerely, your BFFs financial vulture partners!"
In short, if your domain name has "forbidden words" in it and it operates in the same industry as the companies whose words you "steal" it's possible that google doesn't want to have anything (legally) to do with this, so they simply refuse to index your domain name....
Although, now that I think more about it this might not be the case as this would not index any of your URLs. But other URLs do show up, so it's definitely not a trademarking issue... It could still be a DMCA issue, though as - from what I know - DMCA takedowns can happen for individual pages, and this could justify why only the homepage can't be indexed while the other pages are fine...
Anyway, I can't think of anything else that could be wrong, so I'll go with the DMCA option until someone else can figure it out...