def insert_dirichlet_L_functions(start, end):
print "Putting Dirichlet L-functions into database."
start = max(3, start)
for q in range(start, end):
print "Working on modulus", q
sys.stdout.flush()
G = DirichletGroup(q)
for n in range(len(G)):
chi = G[n]
if chi.is_primitive():
L = lc.Lfunction_from_character(chi)
z = L.find_zeros_via_N(1)[0]
Lfunction_data = {}
Lfunction_data['first_zero'] = float(z)
Lfunction_data[
'description'] = "Dirichlet L-function for character number " + str(
n) + " modulo " + str(q)
Lfunction_data['degree'] = 1
Lfunction_data['signature'] = (1, 0)
if chi.is_odd():
Lfunction_data['mu'] = [
(1.0, 0.0),
]
else:
Lfunction_data['mu'] = [
(0.0, 0.0),
]
Lfunction_data['level'] = q
coeffs = []
for k in range(0, 10):
coeffs.append(CC(chi(k)))
Lfunction_data['coeffs'] = [(float(x.real()), float(x.imag()))
for x in coeffs]
Lfunction_data['special'] = {
'type': 'dirichlet',
'modulus': q,
'number': n
}
Lfunctions.insert(Lfunction_data)
import sys
from sage.all import DirichletGroup
import pymongo
from pymongo import Connection
import sage.libs.lcalc.lcalc_Lfunction as lc
from lmfdb import base
C = base.getDBConnection()
db = C.Lfunctions
first_zeros = db.first_zeros_testing
first_zeros.drop()
for q in range(3, 1500):
print q
sys.stdout.flush()
G = DirichletGroup(q)
for n in range(len(G)):
if G[n].is_primitive():
L = lc.Lfunction_from_character(G[n])
z = L.find_zeros_via_N(1)[0]
first_zeros.insert({
'zero': float(z),
'modulus': int(q),
'character': int(n)
})
def __init__(self, **args):
#Check for compulsory arguments
if not ('charactermodulus' in args.keys() and 'characternumber' in args.keys()):
raise KeyError, "You have to supply charactermodulus and characternumber for the L-function of a Dirichlet character"
# Initialize default values
self.numcoeff = 30 # set default to 30 coefficients
# Put the arguments into the object dictionary
self.__dict__.update(args)
self.charactermodulus = int(self.charactermodulus)
self.characternumber = int(self.characternumber)
self.numcoeff = int(self.numcoeff)
# Create the Dirichlet character
chi = DirichletGroup(self.charactermodulus)[self.characternumber]
if chi.is_primitive():
# Extract the L-function information from the Dirichlet character
# Warning: will give nonsense if character is not primitive
aa = int((1-chi(-1))/2) # usually denoted \frak a
self.quasidegree = 1
self.Q_fe = float(sqrt(self.charactermodulus)/sqrt(math.pi))
self.sign = 1/(I**aa * float(sqrt(self.charactermodulus))/(chi.gauss_sum_numerical()))
self.kappa_fe = [0.5]
self.lambda_fe = [0.5*aa]
self.mu_fe = [aa]
self.nu_fe = []
self.langlands = True
self.primitive = True
self.degree = 1
self.level = self.charactermodulus
self.dirichlet_coefficients = []
for n in range(1,self.numcoeff):
self.dirichlet_coefficients.append(chi(n).n())
self.poles = []
self.residues = []
self.coefficient_period = self.charactermodulus
# Determine if the character is real (i.e., if the L-function is selfdual)
chivals=chi.values_on_gens()
self.selfdual = True
for v in chivals:
if abs(imag_part(v)) > 0.0001:
self.selfdual = False
if self.selfdual:
self.coefficient_type = 1
else:
self.coefficient_type = 2
self.texname = "L(s,\\chi)"
self.texnamecompleteds = "\\Lambda(s,\\chi)"
if self.selfdual:
self.texnamecompleted1ms = "\\Lambda(1-s,\\chi)"
else:
self.texnamecompleted1ms = "\\Lambda(1-s,\\overline{\\chi})"
self.credit = 'Sage'
self.citation = ''
self.title = "Dirichlet L-function: $L(s,\\chi)$"
self.title = (self.title+", where $\\chi$ is the character modulo "+
str(self.charactermodulus) + ", number " +
str(self.characternumber))
self.sageLfunction = lc.Lfunction_from_character(chi)
else: #Character not primitive
raise Exception("The dirichlet character you choose is " +
"not primitive so it's Dirichlet series " +
"is not an L-function." ,"UserError")
constructor_logger(self,args)