Completely removing of Site Title and tag line

I want to remove the site title and the tagline completely from my wp site because i modified the theme use a bit changing the logo with the one I made. But what happens now is that when I leave the Site Title and tag line empty in the admin panel every page’s title looks like this “| Page 1”. If i type something in there it does show the site title like “Jimos | Page 1” but it also appears on the logo I’ve put and that doesn’t look good. Is there a way to define the site’s title without having it displayed in the logo’s position though?
Thank you in advance.