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db-Lfunctions.md

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Database Lfunctions
Description L-functions
Status production
Contact Jonathan Bober, David Farmer
Code lfunctions
Collections instances, Lfunctions

Todo: Document indexes for Lfunctions (some appear to be obsolete or never used)

Collection instances

Field type Example
url string 'EllipticCurve/Q/162/d'
Lhash string
type string 'ECQ' or 'G2Q' or 'DIR' or ...

Indexes for instances collection:

  • Lhash: lookup by L-function hash (used to find all objects with the same L-function)
  • url: lookup by object homepage URL
  • type: used to filter instances by type

Collection Lfunctions

Field type Example
origin string 'EllipticCurve/Q/162/d'
load_key string 'cremona'
Lhash string ?
conductor integer 162
degree integer 2
motivic_weight integer 1
gamma_factors a pair of lists of numbers the first list being the Gamma_R shifts, and the second list being the Gamma_C shifts. store, e.g., '1/2' if exact. [[],[0]]
primitive boolean True
root_number number 1
sign_arg number 0
central_character string (conrey label) '162.1'
self_dual bool True
conjugate string (the Lhash) entry of the conjugate
symmetry_type string 'unitary' or 'orthogonal' or 'symplectic'
algebraic bool True
leading_term number 1.736780166943524
values list of pairs of numbers (optional) []
euler_factors list of lists (little endian polynomials) [[1, -1], [1], [1, -3, 5], [1, 4, 7], ...]
bad_lfactors list of pairs (number, list(polynomial)) [[2, [1, -1]], [3, [1]]]
dirichlet_coefficients list of numbers (optional)
st_group string 'SU(2)'
types list of strings ('EC', 'MF', 'DIR', 'G2'...?) ['EC', 'MF']
order_of_vanishing integer 0
coefficient_field string (a field label or description) '1.1.1.1'
analytic_normalization number .5
positive_zeros list of (strictly positive) numbers [...]
plot_delta number (float expected)
plot_values list of numbers (floats expected), which are values of Z(k * plot_delta) for k = 0, 1, 2, ...
z1 float (imaginary part of the first positive zero)
z2 float (imaginary part of the second positive zero)
z3 float (imaginary part of the third positive zero)
a2 pair of floats (a complex number), the embedding of the 2nd dirichlet coefficient in the analytic normalization
a3 pair of floats
... ...
a9 pair of floats
a10 pair of floats
A2 string (integer in arithmetic normalization)
A3 string
... ...
A9 string
A10 string