Recognizing numbers using PARI/GP

[kevin lucas <lucaskevin296@gmail.com>]

I recently had cause to run a computation that spat out the number 2.0298832… A little research suggested that this should be a special value of a Dedekind zeta function, but I can’t find the exact relation. Now, I’m aware that PARI/GP can recognize algebraic numbers using algdep, and one can easily incorporate numbers involving common constants like $\pi$ and e with qflll. But what if you suspect relations between special values of special functions (e.g. eta/gamma/zeta functions) and you don’t know which values of which functions, as in this case? I’ve known mathematicians who found relations like this all the time, which leads me to believe there are some dark arts in PARI for this that are only well known within a small community.

So my questions are:

1. What is this number above?

2. Are there general approaches (or better yet, GP scripts) for finding such relations when one doesn’t know beforehand the constants that will appear? Put simply, how do you recognize numbers in GP?

Any help or reference would be helpful.

Thanks,

Kevin