Lfunction data - 1-1968-1968.509-r0-0-0

Formats: - HTML - YAML - JSON - 2026-07-15T14:55:54.406249

• lfunc_lfunctions •

  Column	Type
  A10	numeric
  A2	numeric
  A3	numeric
  A4	numeric
  A5	numeric
  A6	numeric
  A7	numeric
  A8	numeric
  A9	numeric
  Lhash	text
  a10	jsonb
  a2	jsonb
  a3	jsonb
  a4	jsonb
  a5	jsonb
  a6	jsonb
  a7	jsonb
  a8	jsonb
  a9	jsonb
  accuracy	smallint
  algebraic	boolean
  analytic_conductor	double precision
  analytic_normalization	numeric
  bad_lfactors	jsonb
  bad_primes	bigint[]
  central_character	text
  coeff_info	jsonb
  coefficient_field	text
  conductor	numeric
  conductor_radical	integer
  conjugate	text
  credit	text
  degree	smallint
  dirichlet_coefficients	jsonb
  euler_factors	jsonb
  euler_factors_factorization	jsonb
  gamma_factors	jsonb
  group	text
  index	smallint
  label	text
  leading_term	text
  load_key	text
  motivic_weight	smallint
  mu_imag	numeric[]
  mu_real	smallint[]
  nu_imag	numeric[]
  nu_real_doubled	smallint[]
  order_of_vanishing	smallint
  origin	text
  plot_delta	numeric
  plot_values	jsonb
  positive_zeros	jsonb
  precision	smallint
  prelabel	text
  primitive	boolean
  rational	boolean
  root_analytic_conductor	double precision
  root_angle	double precision
  root_number	text
  self_dual	boolean
  sign_arg	numeric
  spectral_label	text
  st_group	text
  symmetry_type	text
  trace_hash	bigint
  types	jsonb
  values	jsonb
  z1	numeric
  z2	numeric
  z3	numeric

  
  {'Lhash': 'dirichlet_L_1968.509', 'a10': [1.0, -2.8163065144692913e-30], 'a2': [1.0, -2.8163065144692913e-30], 'a3': [1.0, -2.8163065144692913e-30], 'a4': [1.0, -2.8163065144692913e-30], 'a5': [-0.8090169943749475, 0.5877852522924731], 'a6': [1.0, -2.8163065144692913e-30], 'a7': [-0.4539904997395468, -0.8910065241883679], 'a8': [1.0, -2.8163065144692913e-30], 'a9': [1.0, -2.8163065144692913e-30], 'algebraic': True, 'analytic_conductor': 9.139353577353202, 'analytic_normalization': 0, 'bad_lfactors': [[2, [1]], [3, [1]], [41, [1]]], 'bad_primes': [2, 3, 41], 'central_character': '1968.509', 'coeff_info': ['x^40 - 1', '0.987688340595138 + 0.156434465040231*I', '\\\\zeta_{40}'], 'conductor': 1968, 'conductor_radical': 246, 'conjugate': 'dirichlet_L_1968.1013', 'degree': 1, 'dirichlet_coefficients': ['a^0', 0, 0, 0, 'a^56', 0, 'a^27', 0, 0, 0, 'a^69', 0, 'a^33', 0, 0, 0, 'a^29', 0, 'a^27', 0, 0, 0, 'a^28', 0, 'a^32', 0, 0, 0, 'a^21', 0, 'a^4', 0, 0, 0, 'a^43', 0, 'a^46', 0, 0, 0, 0, 0, 'a^48', 0, 0, 0, 'a^13', 0, 'a^14', 0, 0, 0, 'a^61', 0, 'a^45', 0, 0, 0, 'a^58', 0, 'a^12', 0, 0, 0, 'a^9', 0, 'a^11', 0, 0, 0, 'a^39', 0, 'a^30', 0, 0, 0, 'a^16', 0, 'a^35', 0, 0, 0, 'a^30', 0, 'a^45', 0, 0, 0, 'a^47', 0, 'a^60', 0, 0, 0, 'a^3', 0, 'a^21', 0, 0, 0], 'euler_factors': [[0], [0], [1, 'a^56'], [1, 'a^27'], [1, 'a^69'], [1, 'a^33'], [1, 'a^29'], [1, 'a^27'], [1, 'a^28'], [1, 'a^21'], [1, 'a^4'], [1, 'a^46'], [0], [1, 'a^48'], [1, 'a^13'], [1, 'a^61'], [1, 'a^58'], [1, 'a^12'], [1, 'a^11'], [1, 'a^39'], [1, 'a^30'], [1, 'a^35'], [1, 'a^30'], [1, 'a^47'], [1, 'a^21'], [1, 'a^67']], 'gamma_factors': [[0], []], 'index': 0, 'label': '1-1968-1968.509-r0-0-0', 'leading_term': '-0.0328954966184412249817918083531-0.409305382974795513886267033608*I', 'load_key': 'dirichlet', 'motivic_weight': 0, 'mu_imag': [{'__RealLiteral__': 0, 'data': '0.0', 'prec': 10}], 'mu_real': [0], 'nu_imag': [], 'nu_real_doubled': [], 'order_of_vanishing': 0, 'origin': 'Character/Dirichlet/1968/509', 'plot_delta': {'__RealLiteral__': 0, 'data': '0.3125', 'prec': 20}, 'plot_values': [-0.41062514563762176, 0.35625976483360666, 0.8569265250663294, 0.3398429331105133, -1.543008138053377, -3.8998741298204136, -5.078206610261073, -4.107518448313272, -1.7581604837716143, 0.011991182732917795, -0.07649062289544027, -1.2974500466713381, -1.7030904313606516, -0.3839398361246854, 1.2872773159646023, 1.067515645067189, -1.460572669633132, -4.061038951731685, -4.144592383655263, -1.7240010811247908, 0.5286548132885978, 0.3856018842847092, -1.3524821695752776, -2.0347486678440516, -0.67726125215561, 0.7284669429921538, -0.04516515046281113, -2.210277196924376, -2.687491349829382, -0.1473859656697813, 2.9380154348260206, 3.2074649279959546, 0.6414644120267731, -1.528855399708235, -1.1655538445670053, 0.10372218315689061, -0.4318517266701746, -2.3719184925162278, -2.6066613840014283, 0.0899028450421932, 2.897211929053169, 2.6103674166953077, 0.09602185173745478, -0.9263362356733824, 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'87.33477922268555319664635385726', '87.951938680343056299823214418029', '88.61231297276336684566980571523', '89.24990141074771560404712279791', '89.71985412008224593156132408470', '90.30012493402538387431335541461', '91.069220705312046872704826527739', '91.58685761592957111926827828047', '92.24785009092135392827347256747', '92.90429591739586640901708654394', '93.51625769758428163646568161362', '94.13459587585097210227526772709', '94.45268797762818352019690788810', '95.188229210976412160443930429490', '96.08570491549193240917529417618', '96.65628597970392037183665675821', '97.01366161740990422336996971312', '97.76827013493313644421809238219', '98.24627974227801549899905388081', '98.689294239271406139277124078299', '99.87042900345618178534662586522', '100.19227916023482567896239497681', '100.888509523641746049049030881285', '101.083395440810301448033365309519', '101.92201069330943066875416845874', '102.70581826597631753076563675968', '103.05429848349192631238819445229', '103.99267674292240586593637889806', '104.36360214689829253933858403377', '104.9963832123063805710290628024', '105.48387081361807876216240885625', '106.2414119521271271376061022421', '106.776719275171104646299356504367', '107.4598027302916287682142090417', '108.021646702903838131123784667721', '108.79630715172071423660106571685', '109.0089461160284865681202823852', '109.85025063841753752641129191275', '110.42566711846930602960761475297', '111.07288939667291665272201002090', '111.41612989317853316964481549625', '112.48129644709079263850701687183', '112.88711498280787863305549222327', '113.41705226402701304543796513353', '113.85587187834857932401213495819', '114.38262573538702790609495596362', '115.64195862128430309483265504447', '115.847282062803023134821677566096', '116.27213304280415756714104496377', '117.0356556229876758832278732479', '117.538341562896105292732351267488', '118.18885327807290805251755895633', '118.712311304460889743956409434458', '119.59094597615726943160123296154', 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'150.8519722346760220661368381553', '151.244287296918836226047404657511', '151.95891207750684054423062780413', '152.49760792617509376840742832108', '152.78106750807461240822249686843', '153.372972674825651433744823650152', '154.41130286016546352232661243978', '154.88658583300985909378131670653', '155.47816875387543701235778982341', '155.65719602479434416994202818789', '156.53758915598061393391233188483', '156.85837393103836140450391141205', '157.96571666706990185721356976723', '158.2013290000688524856867029875', '158.72912721563330215866847717475', '159.536116502906802805016746819454', '160.05059245492999464815326688273', '160.3791572127954320415419082930', '161.01771173602313310571562718814', '161.90518410358314236538506416886', '162.460986672685174323878904199996', '162.87072530055612750025048540819', '163.2100877746751527360713001884', '164.30260307834477475586346568513', '164.63624445798139260198222060368', '164.96306803441791207408903689480', '165.86634880025382921383483744159', '166.37896035822944659121690418356', '166.956253931395233875039045158660', '167.7128242519386659121161385764', '167.95261150029361874904746110130', '168.6618029884848802819481266324', '169.10274217737640078491142430838', '169.820787427127259182581030448375', '170.64646399885827951760163239716', '171.03866304558163322136586765088', '171.64887231797133670916426444234', '171.802676237622671387684547823922', '172.86209212293879146277692127538', '173.09762967804634939453681759406', '174.01627395339715820805222127767', '174.570337419860126523447818031593', '175.03034928089375785664304689788', '175.61686292335548416011081411054', '175.8905409800941222084192819626', '176.84545343033443282383182602168', '177.29096107019862977145957027394', '178.010281577992320716341582512188', '178.58855362240445650620804350058', '178.97265611215021514161174662300', '179.485815604191289729867899907526', '180.28717448927562661470310931641', '180.61506022212342422681788594628', '181.581912263520180757559579106001', '181.89318029862445909394748857500', '182.36905037051958930517346706250', '183.08238562330073695420323845961', '183.76802491303528474300843696399', '184.20912149266353252200199759819', '184.63558236902046299870366042569', '185.22579931912569942639210036668', '186.18543153905540764548168493896', '186.62885159097571625635297803556', '186.94036653988482583314379305256', '187.64420387716839568915053910231', '188.07991536329152322139075986546', '188.85644612445655971541567599200', '189.30840331107384772832047229055', '190.016132594873122932444192413410', '190.64623284560871848595403719370', '191.08580245977755773032205538394', '191.53646351185380311867464233609', '192.14845997265928062046232952125', '192.608016837011233734092099938984', '193.68854393535578211335801199401', '194.00048751661042797105923680620', '194.49810883636786265711587214196', '194.99413753530009951340374757224', '195.38619665110402230857092356671', '196.40495212093924914472849688922', '196.77424125691872579803121911304', '197.3765690858027822743362768613', '197.95196852360756845298757456720', '198.45739656023883000002979539226', '199.08518155304400064118519157158', '199.594876640818340513915852267184'], 'prelabel': '1-1968-1968.509-r0-0', 'primitive': True, 'rational': False, 'root_analytic_conductor': 9.139353577353202, 'root_angle': 0.47447259540114484, 'root_number': '-0.987164529044368215926122308502+0.159706582821781961272404585697*I', 'self_dual': False, 'sign_arg': {'__RealLiteral__': 0, 'data': '0.474472595401', 'prec': 47}, 'spectral_label': 'r0-0', 'st_group': 'mu(40)', 'symmetry_type': 'unitary', 'types': ['DIR'], 'values': [[1, '0.709017684282277712957493316586-0.194111288398339976951322032208*I']], 'z1': {'__RealLiteral__': 0, 'data': '0.163258195225', 'prec': 47}, 'z2': {'__RealLiteral__': 0, 'data': '1.01264942534', 'prec': 44}, 'z3': {'__RealLiteral__': 0, 'data': '2.80840335387', 'prec': 44}}
• lfunc_search •

  Column	Type
  algebraic	boolean
  analytic_conductor	double precision
  bad_primes	bigint[]
  central_character	text
  conductor	numeric
  conductor_radical	bigint
  degree	smallint
  dirichlet_coefficients	numeric[]
  euler11	numeric[]
  euler13	numeric[]
  euler17	numeric[]
  euler19	numeric[]
  euler2	numeric[]
  euler23	numeric[]
  euler29	numeric[]
  euler3	numeric[]
  euler31	numeric[]
  euler37	numeric[]
  euler41	numeric[]
  euler43	numeric[]
  euler47	numeric[]
  euler5	numeric[]
  euler53	numeric[]
  euler59	numeric[]
  euler61	numeric[]
  euler67	numeric[]
  euler7	numeric[]
  euler71	numeric[]
  euler73	numeric[]
  euler79	numeric[]
  euler83	numeric[]
  euler89	numeric[]
  euler97	numeric[]
  euler_factors	numeric[]
  index	smallint
  instance_types	text[]
  instance_urls	text[]
  is_instance_Artin	boolean
  is_instance_BMF	boolean
  is_instance_CMF	boolean
  is_instance_DIR	boolean
  is_instance_ECNF	boolean
  is_instance_ECQ	boolean
  is_instance_G2Q	boolean
  is_instance_HMF	boolean
  is_instance_MaassGL3	boolean
  is_instance_MaassGL4	boolean
  is_instance_MaassGSp4	boolean
  is_instance_NF	boolean
  label	text
  motivic_weight	smallint
  mu_imag	numeric[]
  mu_real	smallint[]
  nu_imag	numeric[]
  nu_real_doubled	smallint[]
  order_of_vanishing	smallint
  prelabel	text
  primitive	boolean
  rational	boolean
  root_analytic_conductor	double precision
  root_angle	double precision
  self_dual	boolean
  spectral_label	text
  trace_hash	bigint
  z1	numeric

  
  {'algebraic': True, 'analytic_conductor': 9.139353577353202, 'bad_primes': [2, 3, 41], 'central_character': '1968.509', 'conductor': 1968, 'conductor_radical': 246, 'degree': 1, 'index': 0, 'instance_types': ['DIR'], 'instance_urls': ['Character/Dirichlet/1968/509'], 'is_instance_Artin': False, 'is_instance_BMF': False, 'is_instance_CMF': False, 'is_instance_DIR': True, 'is_instance_ECNF': False, 'is_instance_ECQ': False, 'is_instance_G2Q': False, 'is_instance_HMF': False, 'is_instance_MaassGL3': False, 'is_instance_MaassGL4': False, 'is_instance_MaassGSp4': False, 'is_instance_NF': False, 'label': '1-1968-1968.509-r0-0-0', 'motivic_weight': 0, 'mu_imag': [{'__RealLiteral__': 0, 'data': '0.0', 'prec': 10}], 'mu_real': [0], 'nu_imag': [], 'nu_real_doubled': [], 'order_of_vanishing': 0, 'prelabel': '1-1968-1968.509-r0-0', 'primitive': True, 'rational': False, 'root_analytic_conductor': 9.139353577353202, 'root_angle': 0.47447259540114484, 'self_dual': False, 'spectral_label': 'r0-0', 'z1': {'__RealLiteral__': 0, 'data': '0.163258195225', 'prec': 47}}
• lfunc_instances •

  Column	Type
  Lhash	text
  Lhash_array	text[]
  factors	text[]
  label	text
  type	text
  url	text

  
  {'Lhash': 'dirichlet_L_1968.509', 'Lhash_array': ['dirichlet_L_1968.509'], 'factors': ['1-1968-1968.509-r0-0-0'], 'label': '1-1968-1968.509-r0-0-0', 'type': 'DIR', 'url': 'Character/Dirichlet/1968/509'}