# How to find an explicit value of a Hecke L-function using Magma?

I'm trying to compute special values of Hecke L-function for the field $$K=\mathbb{Q}(\sqrt[5]{1})$$ using Magma (more exactly, I need $$L(k, \chi^k)$$, $$k$$ - integer, $$\chi$$ - Hecke character for the field $$K$$). However, I'm very confused, because the text http://magma.maths.usyd.edu.au/~watkins/papers/hecke.pdf says (as far as I understood) that it is possible to do it as we're dealing with a CM field. On the other hand, (again, as far as I got it), one needs to specify real places to create a HeckeCharacterGroup in Magma, but there are no real places for this field? (While looking for the answer found this question Special values of Hecke L-function, but couldn't understand how to do similarly in my case). I feel that I don't realize something simple here.

And one more question on the subject: there are many Dirichlet characters of given modulus, but when we compute Dirichlet L-function for a quadratic field, we take one specific character which is a Kronecker symbol. Is there something similar for Hecke characters in my case?

I'm sorry if my question is somehow very silly but I couldn't find much clear information on the subject. Thank you in advance!

• There seems to be an example for exactly this field $K$ in the Magma documentation at magma.maths.usyd.edu.au/magma/handbook/text/422#4502. Does that help at all? Jan 4, 2021 at 21:11
• thank you! I saw this example, but couldn't understand it :( I think my problem is that I don't really understand which character do I need. But I guess this link will be helpful when I understand the subject better. Jan 8, 2021 at 16:59

HeckeCharacterGroup(I) : RngOrdIdl -> GrpHecke

Alternatively, an empty sequence of places ("[]") should give the same result. However, you will first need to know where your prospective Hecke character is ramified (the ideal $$I$$).
For the second question, the Kronecker character is a specific character corresponding to a quadratic extension of the rationals. Are you dealing with a specific quadratic extension of the 5th cyclotomic field in your case? If so, there is a QuadraticCharacter command in Magma that might help (same documentation), with an example of extending $${\bf Q}(\sqrt{-7})$$ by $$\sqrt{-118-18\sqrt{-7}}$$.
• Thank you! As for the second question, I'm considering the field $\mathbb{Q}(\sqrt[5]{1})$ as a totally imaginary quadratic extension of a quadratic field $\mathbb{Q}(\sqrt{5})$ if that helps. I'll look for this command, thank you very much for your answer! Jan 8, 2021 at 16:50