I don’t think contacting anyone on Linkedin is weird, because that’s what people are on the site for. People that don’t want to be contacted hide their profiles from the general public.
I noticed that you said “hiring manager” in your question, but “recruiter” in the body of the email. Recruiters will generally ignore you if they don’t have a position for you or if they position gets filled before you make it into the process, so following up twice is fine, but I would save the third follow-up (if necessary) for a month from now just to see if they have anything for you. If you don’t hear anything, take it as a no and move on. Recruiters are fair-weathered friends – when they need you, they will treat you like you’re the king of the world, but once they don’t… This is my experience, anyway. Probably different for those with very specific and in-demand skills sets, like developers, etc, since there is always a demand.
As for “hiring managers,” they should appreciate qualified contacts reaching out to them. When I had jobs open, I would post on Linkedin and never received a single message, meaning that I had to wait for the recruiters to send me resumes and they are typically over worked and don’t have the experience to find anyone other than those that have had the same role in the past.