The Savage Model 340 was made in the period 1950-mid 1960 in Chicopee Falls, MA and in the period mid 1960-1985 in Westfield, MA.
There are no records available to give a more exact date, however, if made in the period 1969-1985, your rifle will have a six digit serial number with a letter prefix.
If your rifle was made between 1950 and 1969, it should have the Savage/Stevens date code.
The Savage/Stevens date code is a lightly stamped 1/4" diameter circle located on the receiver or barrel which contains one or two numbers and a letter. The letter is the date code which began in 1949; A = 1949, B = 1950, C = 1951, D = 1952, etc. Note: “O” and “Q” were omitted from the date code.
There are no records available to indicate to whom rifles were originally sold except to state that it was sold to a distributor. Savage did not sell directly to retailers or individuals.