A Kind of Unique Prime Number

Seems a valid solution is

Less than 7 digits and more than 3 digits

ALL digits in the number are Prime numbers-- some repeated.

All individual digits in the number add up to a Prime Number-- whose digits also add up to a Prime number

The last 2 digits of the number are same as the first 2 digits.

And, it exhibits another property described by a hyphenated word in Wikipedia.

As a starting point, the number of digits must be

The first rule tells us that it's

Given that, the sum of the digits must be

If it's a 5-digit number

However at that point I'm out of steam because there's multiple combinations there, and even more for 6 digits.

I have no idea about the "hyphenated word" part, which IMO doesn't fit the style of the rest of the problem. What if that wiki page changes its title in the future? What if another "hyphenated word" is added to wiki, ruining the uniqueness?

Without this part, I just ignored the no-computers tag and found the following list (assuming that less than $7$ digits means strictly less, and the first and last $2$ digits are the same order, i.e. numbers such as $753257$ don't count):

Based on the description, I'd guess the intended answer should be

However a quick google search didn't yield anything for me. Also the number doesn't appear in OEIS.

Thus I put here this result to everyone's reference, which was a comment but has grown too long.

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EDIT: Based on the hint given in the comment, I think it is

Without that hint I would probably never figure out that hyphenated word (and in fact the title of that wiki page is not hyphenated). And of course I cheated in the first place by using a computer :P