It is most likely a model 1900. They where only made from 1900-1910.
I have one with serial #343641 that was made around 1905.
You can find them on Remington's web site.
Value is hard to determine without picture or more info. Most bids start around $400.
They have date code stamps on the barrel which can be looked up on the manufacture date link on the Remington Society of America. --- Update: It's important to note that the barrel on the shotgun MAY not be the barrel that was originally on the shotgun at purchase. Check the serial number with Remington.
Assuming your Model 11 Remington is a shotgun. Remington doesn't date firearms by serial number. Receivers (which have serial numbers) are made during one specific time. Later, at some point the receiver is assembled into a complete gun. Remington, for some reason chooses to use "date codes" that are stamped on the shotgun barrel, so assuming the barrel is the original barrel to that firearm, the date code on the barrel holds the key to when that shotgun became an assembled firearm. Actually serial list/dates do exist ;) They are online at the Remington Society of America in the part of the Remington factory log book including model 11's, from my quick list this is a very late shotgun after 1931 (where my quick serial list ends). There is also the date code table there, please see related links.
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