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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 Description Type Example of stored data Remarks
_id Mongo id ObjectId - assigned by Mongo; contains creation timestamp
origin an LMFDB url from which we have derived this data string u'EllipticCurve/Q/162/d' Not every object has this attribute
load_key a string identifying who uploaded the entry string u'Cremona'
u'costa'
Lhash A string that is used to connect the collection instances and this collection. See below string
  • u'/ModularForm/GL3/Q/Maass/1/1/-13.5965_-4.76468/1.04846245/'
  • u'12896406231020588'
  • u'12896406231020588,1261421830691399758'
  • u''_3835819448407117806631286663382'
`conductor` The conductor of the L-function integer 162
degree The degree of the L-function integer 2
motivic_weight the motivic weight of the L-function 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. stored, e.g., '1/2' if exact. list of integers or string representing rational numbers [[],[0]]
primitive True iff the L-function is primitive boolean True
root_number the root number of the L-function string or an integer or a floating point number 1
sign_arg the argument of the root number floating point number 0
central_character the central character of the L-function using the conrey label conrey label stored as a string '162.1'
self_dual a boolean identifying if the L-function is self dual bool True
`conjugate` The Lhash string of the conjugate string
symmetry_type identifies the symmetry type of the Galois representation associated to L-function string 'unitary' or 'orthogonal' or 'symplectic'
algebraic identifies if the L-function is of an algebraic nature boolean True
leading_term Leading term of the Taylor expansion of the L-function centered at t = 0 on the critical line floating point number 1.736780166943524 Not every object will have this attribute
values ????? list of pairs of numbers [] Optional
euler_factors All the euler factors up to some prime list of lists (little endian polynomials) [[1, -1], [1], [1, -3, 5], [1, 4, 7], ...]
bad_lfactors list all the bad euler factors list of pairs pairs (prime, list(polynomial)) [[2, [1, -1]], [3, [1]]]
dirichlet_coefficients lists of dirichlet coefficients list of numbers ??? Optional
st_group a string representing the Sato-Tate group of to the Galois representation associate to this L-function 'SU(2)'
types ?? list of strings ('EC', 'MF', 'DIR', 'G2'...?) ['EC', 'MF'] Optional??
order_of_vanishing Order of vanishing of the L-function integer 0
coefficient_field The field of coefficients of the Dirichlet coefficients string (a field label or description) '1.1.1.1'
analytic_normalization a floating point number representing how to from the algebraic normalization to analytic normalization floating point number .5
accuracy bit accuracy (after the decimal point) of the nontrivial zeros integer 100 Not optional! However, not all entries in the DB have it at the moment
positive_zeros List of strictly positive zeros stored as strings list of floats [u'2.583212561785407', u'3.900749965053077', u'5.857403178467226', u'7.384090002225106', u'8.414042470938793', u'9.493169657561115', u'10.36534498208387', u'12.03947269145467', u'12.92148359796848', u'14.10055486094281', u'14.62109083140550', u'15.77984643452670', u'16.67808032918220', u'17.84322890580079', u'19.23339240606807', u'19.37702400620986'] Not all digits need to be correct. See accuracy, to get the right number of bits
plot_values list of numbers (floats expected), which are values of Z(k * plot_delta) for k = 0, 1, 2, ... list of floats [1.263098962276239, ...., 0.5305687471043774]
plot_delta the spacing on the x number (float expected) 0.25
z1, z2, z3 imaginary part of the kth positive zero string representing a floating point 5.85740317846722
a2, a3, ..., a10 pair of floats (a complex number), the embedding of the kth Dirichlet coefficient in analytic normalization as a complex number pair of floats u'[0.0, 0,5]'
A2, A3, ..., A10 string representing the kth Dirichlet coefficient in arithmetic normalization string u'2'
Group the codomain group of the galois representation string u'GL2' or u'GL3' or u'GSp4' Not every entry has this attribute

Lhash description

Modular forms

it matches the url of the origin object

genus 2 curves and elliptic curves over QQ

it's an integer stored as a string, the integer is obtained by the Hash function described in Section 4.3 of arXiv 1602.03715

elliptic curves over a numberfield

  • For a primitive L-function we use str(<first zero of L_0> * 2^100).round()).

  • If we do not have enough precision for the integer above to be correct we prepend an underscore.

  • For the nonprimitive case, if the factors are on the database, it's the Lhash of the factors separated by a comma, with no spaces.

  • For last, if one of the primitive factors is not in the database we use str(<first zero of L_0> * 2^100).round()) prepended with an underscore.