strange PBN indexing issue.. any advise?

I bought an expired domain to add to my pbn but the site is not showing on google when you search the full domain (http://www.pbnsite.com)

BUT.. when you search for: www.pbnsite.com OR pbnsite.com its the first result

I got the domain a few months ago and has plenty of content on it (so in other words, its not new)

I was thinking that it may be penalized, thus not showing as first result when searching full url, but if thats the case, why is it showing for the domain searches?

ALSO

If you search the full url for any of the posts, (ie. http://www.pbnsite.com/post1) the page is first result

What do you guys think? Should I drop it from my PBN? or is it nothing to worry about?

Advice anyone?...

Use the site: operator. If it shows up fine, you don't have to worry about a thing. For extra caution, view the site's history on archive.org. Make sure the site that used to exist on the domain wasn't a spam site.

Maybe it was set as https in the past?

If the site comes up when you search the partial URL its clearly indexed. As somebody said before it may have been an HTTPS (which you SHOULD be using anyway) in its past life. I'd switch it over to an HTTPS if it were mine for other reasons, but in your case it seems there may be extra reasons.

thanks for the replies guys. Im not going to bother changing it to a https as I only need it as part of my PBN