# lfunc_search downloaded from the LMFDB on 27 July 2026. # Search link: https://www.lmfdb.org/L/rational/8/75^4 # Query "{'degree': 8, 'conductor': 31640625, 'rational': True}" returned 68 lfunc_searchs, sorted by root analytic conductor. # Each entry in the following data list has the form: # [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, $\epsilon$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "8-75e4-1.1-c1e4-0-0" 0.7738720922475517 0.12863332082565773 8 31640625 "1.1" [] [[0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.51080945135387386955239285171 ["ModularForm/GL2/Q/holomorphic/75/2/e/b"] "8-75e4-1.1-c1e4-0-1" 0.7738720922475517 0.12863332082565773 8 31640625 "1.1" [] [[0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.64009472245150218811766256515 ["ModularForm/GL2/Q/holomorphic/75/2/e/a"] "8-75e4-1.1-c1e4-0-2" 0.7738720922475517 0.12863332082565773 8 31640625 "1.1" [] [[0.5, 0.0], [0.5, 0.0], [0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.81840793571778125165229576876 ["ModularForm/GL2/Q/holomorphic/75/2/g/a"] "8-75e4-1.1-c2e4-0-0" 1.4295460757534983 17.441557395580137 8 31640625 "1.1" [] [[1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.40958377309226192183101428220 ["ModularForm/GL2/Q/holomorphic/75/3/f/a"] "8-75e4-1.1-c2e4-0-1" 1.4295460757534983 17.441557395580137 8 31640625 "1.1" [] [[1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.56232865399743246886736453768 ["ModularForm/GL2/Q/holomorphic/75/3/d/b"] "8-75e4-1.1-c2e4-0-2" 1.4295460757534983 17.441557395580137 8 31640625 "1.1" [] [[1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0]] 2 false 1 0 0.908229494399085101493476265249 ["ModularForm/GL2/Q/holomorphic/75/3/d/c"] "8-75e4-1.1-c2e4-0-3" 1.4295460757534983 17.441557395580137 8 31640625 "1.1" [] [[1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0]] 2 false 1 0 1.00593379324106127071407779195 ["ModularForm/GL2/Q/holomorphic/75/3/f/b"] "8-75e4-1.1-c2e4-0-4" 1.4295460757534983 17.441557395580137 8 31640625 "1.1" [] [[1.0, 0.0], [1.0, 0.0], [1.0, 0.0], [1.0, 0.0]] 2 false 1 0 1.36898646714398648337005601066 ["ModularForm/GL2/Q/holomorphic/75/3/f/c"] "8-75e4-1.1-c3e4-0-0" 2.103602445908102 383.45052511494544 8 31640625 "1.1" [] [[1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.20214257444513072306574759506 ["ModularForm/GL2/Q/holomorphic/75/4/e/a"] "8-75e4-1.1-c3e4-0-1" 2.103602445908102 383.45052511494544 8 31640625 "1.1" [] [[1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.61540808657969934405038730369 ["ModularForm/GL2/Q/holomorphic/75/4/b/c"] "8-75e4-1.1-c3e4-0-2" 2.103602445908102 383.45052511494544 8 31640625 "1.1" [] [[1.5, 0.0], [1.5, 0.0], [1.5, 0.0], [1.5, 0.0]] 3 false 1 0 0.63488197754918008055775487603 ["ModularForm/GL2/Q/holomorphic/75/4/e/b"] "8-75e4-1.1-c4e4-0-0" 2.7843755550012155 3612.6217997006643 8 31640625 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.04835374779150557671626670037 ["ModularForm/GL2/Q/holomorphic/75/5/d/c"] "8-75e4-1.1-c4e4-0-1" 2.7843755550012155 3612.6217997006643 8 31640625 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.05064197751212270300992743704 ["ModularForm/GL2/Q/holomorphic/75/5/f/a"] "8-75e4-1.1-c4e4-0-2" 2.7843755550012155 3612.6217997006643 8 31640625 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.35099594927095634669640289743 ["ModularForm/GL2/Q/holomorphic/75/5/d/b"] "8-75e4-1.1-c4e4-0-3" 2.7843755550012155 3612.6217997006643 8 31640625 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.36264027725873022688477607192 ["ModularForm/GL2/Q/holomorphic/75/5/f/c"] "8-75e4-1.1-c4e4-0-4" 2.7843755550012155 3612.6217997006643 8 31640625 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.46398618199337959240624110096 ["ModularForm/GL2/Q/holomorphic/75/5/c/h"] "8-75e4-1.1-c4e4-0-5" 2.7843755550012155 3612.6217997006643 8 31640625 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.53981027989267550136184883233 ["ModularForm/GL2/Q/holomorphic/75/5/f/b"] "8-75e4-1.1-c4e4-0-6" 2.7843755550012155 3612.6217997006643 8 31640625 "1.1" [] [[2.0, 0.0], [2.0, 0.0], [2.0, 0.0], [2.0, 0.0]] 4 false 1 0 0.78400697358742381036765637018 ["ModularForm/GL2/Q/holomorphic/75/5/f/d"] "8-75e4-1.1-c5e4-0-0" 3.4682540976655027 20935.689300513717 8 31640625 "1.1" [] [[2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.16871843706197433108021374490 ["ModularForm/GL2/Q/holomorphic/75/6/b/e"] "8-75e4-1.1-c5e4-0-1" 3.4682540976655027 20935.689300513717 8 31640625 "1.1" [] [[2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.18689518056172816197187734128 ["ModularForm/GL2/Q/holomorphic/75/6/b/f"] "8-75e4-1.1-c5e4-0-2" 3.4682540976655027 20935.689300513717 8 31640625 "1.1" [] [[2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.46942097395760816159121276717 ["ModularForm/GL2/Q/holomorphic/75/6/e/b"] "8-75e4-1.1-c5e4-0-3" 3.4682540976655027 20935.689300513717 8 31640625 "1.1" [] [[2.5, 0.0], [2.5, 0.0], [2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.55566501642337265539569893758 ["ModularForm/GL2/Q/holomorphic/75/6/e/a"] "8-75e4-1.1-c7e4-0-0" 4.84033851205772 301304.68380712083 8 31640625 "1.1" [] [[3.5, 0.0], [3.5, 0.0], [3.5, 0.0], [3.5, 0.0]] 7 false 1 0 0.07526664663213171612559683120055 ["ModularForm/GL2/Q/holomorphic/75/8/e/a"] "8-75e4-1.1-c7e4-0-1" 4.84033851205772 301304.68380712083 8 31640625 "1.1" [] [[3.5, 0.0], [3.5, 0.0], [3.5, 0.0], [3.5, 0.0]] 7 false 1 0 0.085392835771727571337748015366725 ["ModularForm/GL2/Q/holomorphic/75/8/b/e"] "8-75e4-1.1-c7e4-0-2" 4.84033851205772 301304.68380712083 8 31640625 "1.1" [] [[3.5, 0.0], [3.5, 0.0], [3.5, 0.0], [3.5, 0.0]] 7 false 1 0 0.42101794811626662211555585857262 ["ModularForm/GL2/Q/holomorphic/75/8/b/d"] "8-75e4-1.1-c7e4-0-3" 4.84033851205772 301304.68380712083 8 31640625 "1.1" [] [[3.5, 0.0], [3.5, 0.0], [3.5, 0.0], [3.5, 0.0]] 7 false 1 0 0.51510783714260050662429927666489 ["ModularForm/GL2/Q/holomorphic/75/8/e/b"] "8-75e4-1.1-c7e4-0-4" 4.84033851205772 301304.68380712083 8 31640625 "1.1" [] [[3.5, 0.0], [3.5, 0.0], [3.5, 0.0], [3.5, 0.0]] 7 false 1 0 0.77249770179707580692586814446 ["ModularForm/GL2/Q/holomorphic/75/8/a/j"] "8-75e4-1.1-c7e4-0-5" 4.84033851205772 301304.68380712083 8 31640625 "1.1" [] [[3.5, 0.0], [3.5, 0.0], [3.5, 0.0], [3.5, 0.0]] 7 false 1 4 1.23475592003302044560197847390 ["ModularForm/GL2/Q/holomorphic/75/8/a/i"] "8-75e4-1.1-c8e4-0-0" 5.527512619124858 871440.9054182499 8 31640625 "1.1" [] [[4.0, 0.0], [4.0, 0.0], [4.0, 0.0], [4.0, 0.0]] 8 false 1 0 0.02431373328381196371886089797463 ["ModularForm/GL2/Q/holomorphic/75/9/f/a"] "8-75e4-1.1-c8e4-0-1" 5.527512619124858 871440.9054182499 8 31640625 "1.1" [] [[4.0, 0.0], [4.0, 0.0], [4.0, 0.0], [4.0, 0.0]] 8 false 1 0 0.28752246922820041721980186388087 ["ModularForm/GL2/Q/holomorphic/75/9/d/b"] "8-75e4-1.1-c9e4-0-0" 6.215117674854636 2226357.2115213773 8 31640625 "1.1" [] [[4.5, 0.0], [4.5, 0.0], [4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.093844874986505801546363170257752 ["ModularForm/GL2/Q/holomorphic/75/10/e/b"] "8-75e4-1.1-c9e4-0-1" 6.215117674854636 2226357.2115213773 8 31640625 "1.1" [] [[4.5, 0.0], [4.5, 0.0], [4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.16420147883217560514570593754569 ["ModularForm/GL2/Q/holomorphic/75/10/e/a"] "8-75e4-1.1-c9e4-0-10" 6.215117674854636 2226357.2115213773 8 31640625 "1.1" [] [[4.5, 0.0], [4.5, 0.0], [4.5, 0.0], [4.5, 0.0]] 9 false 1 4 1.45518994833629041348781448288 ["ModularForm/GL2/Q/holomorphic/75/10/a/l"] "8-75e4-1.1-c9e4-0-2" 6.215117674854636 2226357.2115213773 8 31640625 "1.1" [] [[4.5, 0.0], [4.5, 0.0], [4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.17592590938500470586699811319742 ["ModularForm/GL2/Q/holomorphic/75/10/e/c"] "8-75e4-1.1-c9e4-0-3" 6.215117674854636 2226357.2115213773 8 31640625 "1.1" [] [[4.5, 0.0], [4.5, 0.0], [4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.32478098279394818568167629543 ["ModularForm/GL2/Q/holomorphic/75/10/a/i"] "8-75e4-1.1-c9e4-0-4" 6.215117674854636 2226357.2115213773 8 31640625 "1.1" [] [[4.5, 0.0], [4.5, 0.0], [4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.34921553033541188286077896033554 ["ModularForm/GL2/Q/holomorphic/75/10/b/f"] "8-75e4-1.1-c9e4-0-5" 6.215117674854636 2226357.2115213773 8 31640625 "1.1" [] [[4.5, 0.0], [4.5, 0.0], [4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.37685010347195845201505918534564 ["ModularForm/GL2/Q/holomorphic/75/10/e/d"] "8-75e4-1.1-c9e4-0-6" 6.215117674854636 2226357.2115213773 8 31640625 "1.1" [] [[4.5, 0.0], [4.5, 0.0], [4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.39868963713341195395434252827427 ["ModularForm/GL2/Q/holomorphic/75/10/b/e"] "8-75e4-1.1-c9e4-0-7" 6.215117674854636 2226357.2115213773 8 31640625 "1.1" [] [[4.5, 0.0], [4.5, 0.0], [4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.42093737406922370029831749621 ["ModularForm/GL2/Q/holomorphic/75/10/a/k"] "8-75e4-1.1-c9e4-0-8" 6.215117674854636 2226357.2115213773 8 31640625 "1.1" [] [[4.5, 0.0], [4.5, 0.0], [4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.51042537692671486620986065302724 ["ModularForm/GL2/Q/holomorphic/75/10/b/g"] "8-75e4-1.1-c9e4-0-9" 6.215117674854636 2226357.2115213773 8 31640625 "1.1" [] [[4.5, 0.0], [4.5, 0.0], [4.5, 0.0], [4.5, 0.0]] 9 false 1 0 0.895601158618466947108564681988 ["ModularForm/GL2/Q/holomorphic/75/10/a/j"] "8-75e4-1.1-c10e4-0-0" 6.903027882786606 5156048.819730884 8 31640625 "1.1" [] [[5.0, 0.0], [5.0, 0.0], [5.0, 0.0], [5.0, 0.0]] 10 false 1 0 0.05189891337487989931363709660680 ["ModularForm/GL2/Q/holomorphic/75/11/d/b"] "8-75e4-1.1-c11e4-0-0" 7.591161870357474 11027219.456951272 8 31640625 "1.1" [] [[5.5, 0.0], [5.5, 0.0], [5.5, 0.0], [5.5, 0.0]] 11 false 1 0 0.05279509049021667045244243105065 ["ModularForm/GL2/Q/holomorphic/75/12/e/a"] "8-75e4-1.1-c11e4-0-1" 7.591161870357474 11027219.456951272 8 31640625 "1.1" [] [[5.5, 0.0], [5.5, 0.0], [5.5, 0.0], [5.5, 0.0]] 11 false 1 0 0.10662929873570255012677026839 ["ModularForm/GL2/Q/holomorphic/75/12/a/h"] "8-75e4-1.1-c11e4-0-2" 7.591161870357474 11027219.456951272 8 31640625 "1.1" [] [[5.5, 0.0], [5.5, 0.0], [5.5, 0.0], [5.5, 0.0]] 11 false 1 0 0.15624714658301114180173171977249 ["ModularForm/GL2/Q/holomorphic/75/12/b/d"] "8-75e4-1.1-c11e4-0-3" 7.591161870357474 11027219.456951272 8 31640625 "1.1" [] [[5.5, 0.0], [5.5, 0.0], [5.5, 0.0], [5.5, 0.0]] 11 false 1 0 0.24972117325237790715009141799624 ["ModularForm/GL2/Q/holomorphic/75/12/e/b"] "8-75e4-1.1-c11e4-0-4" 7.591161870357474 11027219.456951272 8 31640625 "1.1" [] [[5.5, 0.0], [5.5, 0.0], [5.5, 0.0], [5.5, 0.0]] 11 false 1 0 0.43420913448264102988323528447 ["ModularForm/GL2/Q/holomorphic/75/12/a/i"] "8-75e4-1.1-c11e4-0-5" 7.591161870357474 11027219.456951272 8 31640625 "1.1" [] [[5.5, 0.0], [5.5, 0.0], [5.5, 0.0], [5.5, 0.0]] 11 false 1 0 0.44942137932042965092661217139914 ["ModularForm/GL2/Q/holomorphic/75/12/b/c"] "8-75e4-1.1-c12e4-0-0" 8.279464734855837 22080966.831431784 8 31640625 "1.1" [] [[6.0, 0.0], [6.0, 0.0], [6.0, 0.0], [6.0, 0.0]] 12 false 1 0 0.23339234915815577210636806543 ["ModularForm/GL2/Q/holomorphic/75/13/d/b"] "8-75e4-1.1-c13e4-0-0" 8.967898121295455 41833608.42146173 8 31640625 "1.1" [] [[6.5, 0.0], [6.5, 0.0], [6.5, 0.0], [6.5, 0.0]] 13 false 1 0 0.06307403620646684077692587361 ["ModularForm/GL2/Q/holomorphic/75/14/b/e"] "8-75e4-1.1-c13e4-0-1" 8.967898121295455 41833608.42146173 8 31640625 "1.1" [] [[6.5, 0.0], [6.5, 0.0], [6.5, 0.0], [6.5, 0.0]] 13 false 1 0 0.097220798341164762005522819136 ["ModularForm/GL2/Q/holomorphic/75/14/b/d"] "8-75e4-1.1-c13e4-0-2" 8.967898121295455 41833608.42146173 8 31640625 "1.1" [] [[6.5, 0.0], [6.5, 0.0], [6.5, 0.0], [6.5, 0.0]] 13 false 1 0 0.19582934546571657711785476552 ["ModularForm/GL2/Q/holomorphic/75/14/b/c"] "8-75e4-1.1-c13e4-0-3" 8.967898121295455 41833608.42146173 8 31640625 "1.1" [] [[6.5, 0.0], [6.5, 0.0], [6.5, 0.0], [6.5, 0.0]] 13 false 1 4 0.936857817388879330127202850637 ["ModularForm/GL2/Q/holomorphic/75/14/a/g"] "8-75e4-1.1-c13e4-0-4" 8.967898121295455 41833608.42146173 8 31640625 "1.1" [] [[6.5, 0.0], [6.5, 0.0], [6.5, 0.0], [6.5, 0.0]] 13 false 1 4 1.24460611912974758342472997794 ["ModularForm/GL2/Q/holomorphic/75/14/a/h"] "8-75e4-1.1-c14e4-0-0" 9.656434441030992 75602189.08181262 8 31640625 "1.1" [] [[7.0, 0.0], [7.0, 0.0], [7.0, 0.0], [7.0, 0.0]] 14 false 1 0 0.21667190781384576252175566365 ["ModularForm/GL2/Q/holomorphic/75/15/c/d"] "8-75e4-1.1-c15e4-0-0" 10.34505335051204 131178263.54057658 8 31640625 "1.1" [] [[7.5, 0.0], [7.5, 0.0], [7.5, 0.0], [7.5, 0.0]] 15 false 1 0 0.03033676210177617695266714187 ["ModularForm/GL2/Q/holomorphic/75/16/b/c"] "8-75e4-1.1-c15e4-0-1" 10.34505335051204 131178263.54057658 8 31640625 "1.1" [] [[7.5, 0.0], [7.5, 0.0], [7.5, 0.0], [7.5, 0.0]] 15 false 1 0 0.33299609039243479823310512900 ["ModularForm/GL2/Q/holomorphic/75/16/b/d"] "8-75e4-1.1-c15e4-0-2" 10.34505335051204 131178263.54057658 8 31640625 "1.1" [] [[7.5, 0.0], [7.5, 0.0], [7.5, 0.0], [7.5, 0.0]] 15 false 1 4 0.899471559572655204452510181980 ["ModularForm/GL2/Q/holomorphic/75/16/a/h"] "8-75e4-1.1-c15e4-0-3" 10.34505335051204 131178263.54057658 8 31640625 "1.1" [] [[7.5, 0.0], [7.5, 0.0], [7.5, 0.0], [7.5, 0.0]] 15 false 1 4 1.03439051258657008210104248526 ["ModularForm/GL2/Q/holomorphic/75/16/a/g"] "8-75e4-1.1-c16e4-0-0" 11.033739524407414 219675598.4305879 8 31640625 "1.1" [] [[8.0, 0.0], [8.0, 0.0], [8.0, 0.0], [8.0, 0.0]] 16 false 1 0 0.22230129792491773614443717165 ["ModularForm/GL2/Q/holomorphic/75/17/c/d"] "8-75e4-1.1-c17e4-0-0" 11.722481201026508 356579490.07116103 8 31640625 "1.1" [] [[8.5, 0.0], [8.5, 0.0], [8.5, 0.0], [8.5, 0.0]] 17 false 1 0 0.05873915200154127163503481477 ["ModularForm/GL2/Q/holomorphic/75/18/b/c"] "8-75e4-1.1-c17e4-0-1" 11.722481201026508 356579490.07116103 8 31640625 "1.1" [] [[8.5, 0.0], [8.5, 0.0], [8.5, 0.0], [8.5, 0.0]] 17 false 1 0 0.24157148347029332841601968967 ["ModularForm/GL2/Q/holomorphic/75/18/b/b"] "8-75e4-1.1-c17e4-0-2" 11.722481201026508 356579490.07116103 8 31640625 "1.1" [] [[8.5, 0.0], [8.5, 0.0], [8.5, 0.0], [8.5, 0.0]] 17 false 1 0 0.29184656463571752395965434864 ["ModularForm/GL2/Q/holomorphic/75/18/a/f"] "8-75e4-1.1-c19e4-0-0" 13.10009627511514 867353028.125842 8 31640625 "1.1" [] [[9.5, 0.0], [9.5, 0.0], [9.5, 0.0], [9.5, 0.0]] 19 false 1 0 0.15412655344105688852863806763 ["ModularForm/GL2/Q/holomorphic/75/20/b/c"] "8-75e4-1.1-c19e4-0-1" 13.10009627511514 867353028.125842 8 31640625 "1.1" [] [[9.5, 0.0], [9.5, 0.0], [9.5, 0.0], [9.5, 0.0]] 19 false 1 0 0.17905940700087798304141788972 ["ModularForm/GL2/Q/holomorphic/75/20/b/b"] "8-75e4-1.1-c19e4-0-2" 13.10009627511514 867353028.125842 8 31640625 "1.1" [] [[9.5, 0.0], [9.5, 0.0], [9.5, 0.0], [9.5, 0.0]] 19 false 1 0 0.19197901860509408821221773618 ["ModularForm/GL2/Q/holomorphic/75/20/a/f"] "8-75e4-1.1-c21e4-0-0" 14.477845427248372 1930329663.545941 8 31640625 "1.1" [] [[10.5, 0.0], [10.5, 0.0], [10.5, 0.0], [10.5, 0.0]] 21 false 1 0 0.13215812560812576347308888797 ["ModularForm/GL2/Q/holomorphic/75/22/a/h"] "8-75e4-1.1-c21e4-0-1" 14.477845427248372 1930329663.545941 8 31640625 "1.1" [] [[10.5, 0.0], [10.5, 0.0], [10.5, 0.0], [10.5, 0.0]] 21 false 1 0 0.40058697550043657138456475694 ["ModularForm/GL2/Q/holomorphic/75/22/b/d"] # Label -- # Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where # * $d$ is the degree of $L$. # * $N$ is the conductor of $L$. When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions. # * q.k is the label of the primitive Dirichlet character from which the central character is induced. # * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation. # * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label. #$\alpha$ (root_analytic_conductor) -- # If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$. It plays a role analogous to the root discriminant for number fields. #$A$ (analytic_conductor) -- # The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number # \[ # A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N. # \] #$d$ (degree) -- # The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor. #$N$ (conductor) -- # The **conductor** of an L-function is the integer $N$ occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function # associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$. # In the literature, the word _level_ is sometimes used instead of _conductor_. #$\chi$ (central_character) -- # An L-function has an Euler product of the form # $L(s) = \prod_p L_p(p^{-s})^{-1}$ # where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function. # Here, $N$ is the conductor of $L$. #$\mu$ (mus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$\nu$ (nus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$w$ (motivic_weight) -- # The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$. # If the L-function arises from a motive, then the weight of the motive has the # same parity as the motivic weight of the L-function, but the weight of the motive # could be larger. This apparent discrepancy comes from the fact that a Tate twist # increases the weight of the motive. This corresponds to the change of variables # $s \mapsto s + j$ in the L-function of the motive. #prim (primitive) -- # An L-function is primitive if it cannot be written as a product of nontrivial L-functions. The "trivial L-function" is the constant function $1$. #$\epsilon$ (root_number) -- # The **sign** of the functional equation of an analytic L-function, also called the **root number**, is the complex number $\varepsilon$ that appears in the functional equation of $\Lambda(s)=\varepsilon \overline{\Lambda}(1-s)$. The sign appears as the 4th entry in the quadruple # known as the Selberg data. #$r$ (order_of_vanishing) -- # The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point. # When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point). #First zero (z1) -- # The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$. # Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization). # The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number. #Origin (instance_urls) -- # L-functions arise from many different sources. Already in degree 2 we have examples of # L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions). # Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).