There may be damage to the wiring of line 2 (either in the building or on the phone). Are both lines wired to one jack or are they separate? Do you have any other jacks that have line 2 connected (preferably as single line jack) in the building?
I like to use an old single line corded phone that does not use a power adapter. (As long as the EP5962 corded base will work without power (POTS), it might work for the tests below. You might want a simpler phone to carry around.)
If the lines have separate jacks, try switching line 1 and line 2. (Connect your Line 2 jack to the Line 1 input on the phone and vice versa.) If line 2 doesn't give a dial tone when connected to line 1, see the next set of tests. Otherwise, contact AT&T if the phone is under warranty. These phones are not meant to be user-serviced. Alternatively, plug in a different phone to the line 2 jack to test for dial tone.
Next move to another jack with line 2. Check if you have dial tone at that point. (If you do, you know the problem is at the first jack or the wiring to the jack.)
If there is no dial tone at the next jack or you do not have another jack for the line, find where your telephone service comes in. Open the service box and determine the type of connections in the box. If you are lucky, you will find RJ-11 jacks for the incoming service. Plug in your test phone and check for dial tone. If you have no dial tone or can't check the service, then contact your telephone company. They will test your line and fix it free if the probIem is outside.
If you have a dial tone, then it's an inside wiring problem. Inspect the visible wires for damage. Open the faceplates for the jacks and check for loose wires. There are two types of connectors. Screw type connectors need the wire stripped back a little then wrap around the screw and tighten. The other type is a punchdown jack. To repair a loose wire in this, you will need a punchdown tool. These are often available in hardware/home improvement stores in the wiring section. http://www.lanshack.com/wire_phone_jack.aspx
has clear pictures of the wire color to pin pattern.
You may want a professional to check the wires inside the walls. Your phone company will fix problems for a fee.
Damaged DSL filters on one line could be the cause. Similarly certain animals may have chewed on the specific wires somewhere in the line.
I hope this helps.