A question about L-function data

[Detchat Samart]

Dear LMFDB support team,

I have been working on a certain problem which requires factorization of L-functions of elliptic curves defined over number fields of small degree. In many cases, I found LMFDB data very useful. For example, according to data shown on LMFDB, the L-function https://www.lmfdb.org/L/4/6e4/1.1/c1e2/0/0 corresponding to a CM elliptic curve defined over Q(\sqrt{3}) is the product of two L-functions of weight 2 cusp forms of level 36. 

My question is: how does LMFDB obtain these "origins of factors" data computationally? I found from my computational results that there are many other elliptic curves (both CM and non-CM) whose L-functions appear to be products of those of cusp forms but no similar data are available on LMFDB, so I would like to know how to compute them in general. Any guidance or references regarding this question would be highly appreciated. 

Best regards,

Detchat

[David W. Farmer]

I don't know how the database with that information was populated,
but for L-functions such as the one you mentioned, it is a relatively
straightforward computation to determine if it factors.

By Strong multiplicity one, there is a unique way to write
the functional equation of an L-function. Therefore, given
a degree 4 L-function, one an immediately list the possible
funcitonal equations of any facrots.
Then one just looks up whether there are L-functions
with those functional equations, and then checks if any of those
are indeed factors (by multiplying out the supposed factors
and seeing if the Dirichlet series agree, for example. Or checking
the first few zeros if not all of the possible factors are
available -- which can occur if the conductor is large).

This only works if you have a sufficiently large database
of lower degree L-functions.

Apologies if I am explaining something you already know,
instead of answering your question.

Regards,

David

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[Edgar Costa]

Dear Detchat,

The matching of origins and related objects is done via the first zeros.

For the degree 4 L-function associated with the restriction of scalars of an elliptic curve E over a quadratic field K, we know how they can factor.

- if E is a Q-curve, i.e., E is isogenous to its Galois conjugate, then L(res_K/Q E, s) = L(f,s) L(sigma(f), s), where f is a classical cuspform with an inner twist

- otherwise, L(res_K/Q E, s) = L(f,s) where f is a Hilbert/Bianchi cuspform, and the L-function is primitive.

Best,

Edgar

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