I'd say since functions as an adverb, modifying long, in your sentence.
Here is what the American Heritage Dictionary says:
since adv. 1. From then until now or between then and now: They left town and haven't been here since. 2. Before now; ago: a name long since forgotten. 3. After some point in the past; at a subsequent time: My friend has since married and moved to California.
Thanks to PromegaX for the dictionary information.
The idiomatic phrase "long since" is equivalent to "a long time ago" or, with a past perfect or future perfect verb, "a long time previously." With stative or other durative verbs, it means "for a long time" (see below). The examples below are from Google.
Here's an instance from Google of its use alone, without a verb or even a sentence:
"” Berkowitz: This experiment is presumably over now, right? "” Newhouse: Yes. Long since. There's a Harvard University Press book that summarizes it, as well as a lot of journal articles.
I don't think you can assign a part-of-speech name to the word "since" in the phrase. As an idiomatic adverbial set phrase, it's hard to analyze in terms of its constituent words. In fact, native English speakers have begun to use the phrase as one word, as demonstrated by the 1,450 examples of the single word form that I found on Google. This usage is not condoned yet by dictionaries, but the trend is there nevertheless:
"” These visions from the fifties have longsince given way to sober reality. Besides the military uses of nuclear power, only power stations have proved feasible.
"” Both movies have plenty of charm, and, as I say, reflect an innocence we seem to have longsince lost.
"” As soon as a supply of English currency arrived in the Island [of Jersey], the German Occupation Marks, longsince the only currency in use, was withdrawn and reimbursed at the occupation rate of exchange "”9.36 to the £.
Another curious feature of this phrase is that, whereas with "a long time ago," you have to use the simple past, with "long since" you use one of the perfects to achieve the same meaning. Here are a few examples from Google:
"”In terms of its rural setting and its focus on a small community that, even in Ireland, has long since ceased to exist, the book has real echoes of Dancing at Lughnasa.
"” It was purchased in a store that has long since gone out of business.
It was a marvel how they kept their powder dry. Most of Fort Necessity's defenders had long since found their cartridge boxes soaked, their firearms useless; even the powder under cover in the stockade was wet.
With the future perfect, it means "a long time previously":
"” Airlines aren't much more reliable, and by the time that you locate your fish and have them shipped back to you they will have long since died.
The phrase is used with stative verbs with the meaning "for a long time":
"” "I have long since believed that in spite of all the vigilance that can be infused into post commanders, the specie regulations of the Treasury Department have been violated....
"” We have long since known that the incidence of breast cancer and other female reproductive cancers is at least 75% lower in China and Japan, where soy products are consumed on a daily basis.
"” Doctors have long since suspected a link between smoking and infertility but until now, there has been no scientific research to support the theory.
"” The Bill is welcome. We have long since needed a complete review of valuations nation wide and I am pleased this will take place over the next few years.
"” Although the monastery is long since gone , its name is still preserved in the lovely park southeast of Brussels called "Bois de la Chambre" ("Chamber Woods").
The best way to see the phrase is as an unanalyzable idiomatic adverbial.
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