# lfunc_search downloaded from the LMFDB on 25 August 2026. # Search link: https://www.lmfdb.org/L/2/162/27.13/c1-0 # Query "{'degree': 2, 'conductor': 162, 'spectral_label': 'c1-0'}" returned 192 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, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "2-162-1.1-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.60693251106342657124988453680 ["EllipticCurve/Q/162/b", "ModularForm/GL2/Q/holomorphic/162/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/162/2/a/b"] "2-162-1.1-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 2.19879385611801234222671009191 ["EllipticCurve/Q/162/c", "ModularForm/GL2/Q/holomorphic/162/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/162/2/a/c"] "2-162-1.1-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 2.30016225881827778465495235815 ["EllipticCurve/Q/162/d", "ModularForm/GL2/Q/holomorphic/162/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/162/2/a/d"] "2-162-1.1-c1-0-3" 1.1373550513119213 1.293576512744743 2 162 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 2.94651413139897301734292877776 ["EllipticCurve/Q/162/a", "ModularForm/GL2/Q/holomorphic/162/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/162/2/a/a"] "2-162-27.13-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "27.13" [] [[0.5, 0.0]] 1 true true false false -0.2037037037037037 0 1.52873340963037104889923350361 ["ModularForm/GL2/Q/holomorphic/162/2/e/a/91/1"] "2-162-27.13-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "27.13" [] [[0.5, 0.0]] 1 true true false false 0.04823409676849818 0 1.70213180785438036846964555543 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/91/2"] "2-162-27.13-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "27.13" [] [[0.5, 0.0]] 1 true true false false 0.43324738471298335 0 3.12484185828243026803511606807 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/91/1"] "2-162-27.16-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "27.16" [] [[0.5, 0.0]] 1 true true false false -0.02843136433597984 0 1.40546970735566747806752543355 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/127/2"] "2-162-27.16-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "27.16" [] [[0.5, 0.0]] 1 true true false false -0.12962962962962962 0 1.85363419930606385773560585482 ["ModularForm/GL2/Q/holomorphic/162/2/e/a/127/1"] "2-162-27.16-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "27.16" [] [[0.5, 0.0]] 1 true true false false 0.21361654952116504 0 2.12175435445863802592914378363 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/127/1"] "2-162-27.22-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "27.22" [] [[0.5, 0.0]] 1 true true false false -0.21361654952116504 0 1.13696009375944745073072174369 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/37/1"] "2-162-27.22-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "27.22" [] [[0.5, 0.0]] 1 true true false false 0.02843136433597984 0 2.02873927601914779256361266137 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/37/2"] "2-162-27.22-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "27.22" [] [[0.5, 0.0]] 1 true true false false 0.12962962962962962 0 2.45927333243607376973869572282 ["ModularForm/GL2/Q/holomorphic/162/2/e/a/37/1"] "2-162-27.25-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "27.25" [] [[0.5, 0.0]] 1 true true false false -0.43324738471298335 0 0.62000674302271946184457691417 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/73/1"] "2-162-27.25-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "27.25" [] [[0.5, 0.0]] 1 true true false false -0.04823409676849818 0 2.08072822401724539267379905701 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/73/2"] "2-162-27.25-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "27.25" [] [[0.5, 0.0]] 1 true true false false 0.2037037037037037 0 2.36596287690234497643672623157 ["ModularForm/GL2/Q/holomorphic/162/2/e/a/73/1"] "2-162-27.4-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "27.4" [] [[0.5, 0.0]] 1 true true false false -0.2037037037037037 0 0.892424071567505690977637293974 ["ModularForm/GL2/Q/holomorphic/162/2/e/a/145/1"] "2-162-27.4-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "27.4" [] [[0.5, 0.0]] 1 true true false false -0.155863608098189 0 1.63739275258173341701393103055 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/145/1"] "2-162-27.4-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "27.4" [] [[0.5, 0.0]] 1 true true false false -0.02932157708699618 0 2.21131508332172688628203670088 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/145/2"] "2-162-27.7-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "27.7" [] [[0.5, 0.0]] 1 true true false false 0.02932157708699618 0 2.14794964920307706180648382101 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/19/2"] "2-162-27.7-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "27.7" [] [[0.5, 0.0]] 1 true true false false 0.2037037037037037 0 2.36715758806242702386775584580 ["ModularForm/GL2/Q/holomorphic/162/2/e/a/19/1"] "2-162-27.7-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "27.7" [] [[0.5, 0.0]] 1 true true false false 0.155863608098189 0 2.88472112685116412134947135781 ["ModularForm/GL2/Q/holomorphic/162/2/e/b/19/1"] "2-162-81.13-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "81.13" [] [[0.5, 0.0]] 1 true true false false -0.005143616936720574 0 1.22191347438888706362493032815 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/13/1"] "2-162-81.13-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "81.13" [] [[0.5, 0.0]] 1 true true false false -0.1742103684381298 0 1.22381005831425234021513576589 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/13/4"] "2-162-81.13-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "81.13" [] [[0.5, 0.0]] 1 true true false false -0.1489564883737851 0 1.45462424636730809525512148615 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/13/1"] "2-162-81.13-c1-0-3" 1.1373550513119213 1.293576512744743 2 162 "81.13" [] [[0.5, 0.0]] 1 true true false false -0.22168398538019987 0 1.52197809693834819267696396075 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/13/3"] "2-162-81.13-c1-0-4" 1.1373550513119213 1.293576512744743 2 162 "81.13" [] [[0.5, 0.0]] 1 true true false false -0.0951897461389558 0 2.33405441117148183421818350509 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/13/5"] "2-162-81.13-c1-0-5" 1.1373550513119213 1.293576512744743 2 162 "81.13" [] [[0.5, 0.0]] 1 true true false false -0.016493816345187474 0 2.34727298127201137508406041905 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/13/4"] "2-162-81.13-c1-0-6" 1.1373550513119213 1.293576512744743 2 162 "81.13" [] [[0.5, 0.0]] 1 true true false false 0.472449605047131 0 2.70153864666891934332298071249 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/13/2"] "2-162-81.13-c1-0-7" 1.1373550513119213 1.293576512744743 2 162 "81.13" [] [[0.5, 0.0]] 1 true true false false 0.3241883309450034 0 2.74361528887922098859392503470 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/13/3"] "2-162-81.13-c1-0-8" 1.1373550513119213 1.293576512744743 2 162 "81.13" [] [[0.5, 0.0]] 1 true true false false 0.19837341895417768 0 3.07235917575760619252547212669 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/13/2"] "2-162-81.16-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "81.16" [] [[0.5, 0.0]] 1 true true false false -0.41129364665588297 0 0.49480758355865515257774302174 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/97/2"] "2-162-81.16-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "81.16" [] [[0.5, 0.0]] 1 true true false false -0.0928099838949839 0 0.957095999527916182398819869034 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/97/1"] "2-162-81.16-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "81.16" [] [[0.5, 0.0]] 1 true true false false -0.21524289877335703 0 1.20014358793692803879988970430 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/97/1"] "2-162-81.16-c1-0-3" 1.1373550513119213 1.293576512744743 2 162 "81.16" [] [[0.5, 0.0]] 1 true true false false -0.20590756410538563 0 1.61157395946751076879537954751 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/97/3"] "2-162-81.16-c1-0-4" 1.1373550513119213 1.293576512744743 2 162 "81.16" [] [[0.5, 0.0]] 1 true true false false 0.08322814442258937 0 2.03934956695346106728769090903 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/97/4"] "2-162-81.16-c1-0-5" 1.1373550513119213 1.293576512744743 2 162 "81.16" [] [[0.5, 0.0]] 1 true true false false 0.025813757733215762 0 2.06121770578590649417626780896 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/97/3"] "2-162-81.16-c1-0-6" 1.1373550513119213 1.293576512744743 2 162 "81.16" [] [[0.5, 0.0]] 1 true true false false 0.11233761505964046 0 2.49200665585174521493340495172 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/97/2"] "2-162-81.16-c1-0-7" 1.1373550513119213 1.293576512744743 2 162 "81.16" [] [[0.5, 0.0]] 1 true true false false 0.06562968881682156 0 2.56197349468275972828303079695 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/97/4"] "2-162-81.16-c1-0-8" 1.1373550513119213 1.293576512744743 2 162 "81.16" [] [[0.5, 0.0]] 1 true true false false -0.028421779269324363 0 2.76745412678928797247351510067 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/97/5"] "2-162-81.22-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "81.22" [] [[0.5, 0.0]] 1 true true false false 0.4918762349964357 0 0.31988272978699971607451532457 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/103/1"] "2-162-81.22-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "81.22" [] [[0.5, 0.0]] 1 true true false false -0.196090362991975 0 0.982492822554157887070919041992 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/103/4"] "2-162-81.22-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "81.22" [] [[0.5, 0.0]] 1 true true false false 0.016436025674208297 0 1.27302096723230869400929266977 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/103/1"] "2-162-81.22-c1-0-3" 1.1373550513119213 1.293576512744743 2 162 "81.22" [] [[0.5, 0.0]] 1 true true false false -0.25264070395171373 0 1.61203687055671515347359088388 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/103/2"] "2-162-81.22-c1-0-4" 1.1373550513119213 1.293576512744743 2 162 "81.22" [] [[0.5, 0.0]] 1 true true false false -0.16819337432422632 0 1.89490617403707246793688727530 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/103/4"] "2-162-81.22-c1-0-5" 1.1373550513119213 1.293576512744743 2 162 "81.22" [] [[0.5, 0.0]] 1 true true false false 0.10323869995114937 0 2.14775614097493780472675474139 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/103/5"] "2-162-81.22-c1-0-6" 1.1373550513119213 1.293576512744743 2 162 "81.22" [] [[0.5, 0.0]] 1 true true false false 0.21855281181019534 0 2.23267783575360868593342997522 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/103/2"] "2-162-81.22-c1-0-7" 1.1373550513119213 1.293576512744743 2 162 "81.22" [] [[0.5, 0.0]] 1 true true false false -0.00931376165876726 0 2.40284642529944506437164152207 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/103/3"] "2-162-81.22-c1-0-8" 1.1373550513119213 1.293576512744743 2 162 "81.22" [] [[0.5, 0.0]] 1 true true false false 0.4628010971613603 0 3.24971551875167577500708680394 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/103/3"] "2-162-81.25-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "81.25" [] [[0.5, 0.0]] 1 true true false false -0.472449605047131 0 0.36359044079345667058916013083 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/25/2"] "2-162-81.25-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "81.25" [] [[0.5, 0.0]] 1 true true false false -0.3241883309450034 0 1.06541587930599642142530118216 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/25/3"] "2-162-81.25-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "81.25" [] [[0.5, 0.0]] 1 true true false false -0.19837341895417768 0 1.19906959414094080771113416884 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/25/2"] "2-162-81.25-c1-0-3" 1.1373550513119213 1.293576512744743 2 162 "81.25" [] [[0.5, 0.0]] 1 true true false false 0.005143616936720574 0 1.44494644829390451379095448270 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/25/1"] "2-162-81.25-c1-0-4" 1.1373550513119213 1.293576512744743 2 162 "81.25" [] [[0.5, 0.0]] 1 true true false false 0.1489564883737851 0 2.40213236492239001614897502792 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/25/1"] "2-162-81.25-c1-0-5" 1.1373550513119213 1.293576512744743 2 162 "81.25" [] [[0.5, 0.0]] 1 true true false false 0.1742103684381298 0 2.40994765852084329159636761783 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/25/4"] "2-162-81.25-c1-0-6" 1.1373550513119213 1.293576512744743 2 162 "81.25" [] [[0.5, 0.0]] 1 true true false false 0.016493816345187474 0 2.42906702094086306517380386254 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/25/4"] "2-162-81.25-c1-0-7" 1.1373550513119213 1.293576512744743 2 162 "81.25" [] [[0.5, 0.0]] 1 true true false false 0.0951897461389558 0 2.63315780385185401385809354712 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/25/5"] "2-162-81.25-c1-0-8" 1.1373550513119213 1.293576512744743 2 162 "81.25" [] [[0.5, 0.0]] 1 true true false false 0.22168398538019987 0 2.80942776203676705726141428476 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/25/3"] "2-162-81.31-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "81.31" [] [[0.5, 0.0]] 1 true true false false -0.24581105967290887 0 0.38705065864202513550045584385 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/31/2"] "2-162-81.31-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "81.31" [] [[0.5, 0.0]] 1 true true false false -0.42460264171704976 0 0.74748992836158700644494109421 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/31/1"] "2-162-81.31-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "81.31" [] [[0.5, 0.0]] 1 true true false false -0.11557814827796131 0 1.30237706659071845590454987481 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/31/3"] "2-162-81.31-c1-0-3" 1.1373550513119213 1.293576512744743 2 162 "81.31" [] [[0.5, 0.0]] 1 true true false false -0.33087326263535005 0 1.32600812200688212768983322346 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/31/4"] "2-162-81.31-c1-0-4" 1.1373550513119213 1.293576512744743 2 162 "81.31" [] [[0.5, 0.0]] 1 true true false false 0.07610460012049865 0 1.48277496389866971168036475999 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/31/1"] "2-162-81.31-c1-0-5" 1.1373550513119213 1.293576512744743 2 162 "81.31" [] [[0.5, 0.0]] 1 true true false false -0.13188970562545294 0 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["ModularForm/GL2/Q/holomorphic/162/2/g/a/79/3"] "2-162-81.79-c1-0-4" 1.1373550513119213 1.293576512744743 2 162 "81.79" [] [[0.5, 0.0]] 1 true true false false -0.016197717013279354 0 2.03748952165427069923831334784 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/79/4"] "2-162-81.79-c1-0-5" 1.1373550513119213 1.293576512744743 2 162 "81.79" [] [[0.5, 0.0]] 1 true true false false -0.13916608627736796 0 2.06715752688213658656161254256 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/79/5"] "2-162-81.79-c1-0-6" 1.1373550513119213 1.293576512744743 2 162 "81.79" [] [[0.5, 0.0]] 1 true true false false 0.18492256275677732 0 2.60542591470801487181703882366 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/79/4"] "2-162-81.79-c1-0-7" 1.1373550513119213 1.293576512744743 2 162 "81.79" [] [[0.5, 0.0]] 1 true true false false 0.27513583349195186 0 2.71148811355717911505948240685 ["ModularForm/GL2/Q/holomorphic/162/2/g/a/79/2"] "2-162-81.79-c1-0-8" 1.1373550513119213 1.293576512744743 2 162 "81.79" [] [[0.5, 0.0]] 1 true true false false 0.3239252804460747 0 2.82176566258667111590726922631 ["ModularForm/GL2/Q/holomorphic/162/2/g/b/79/2"] "2-162-9.4-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "9.4" [] [[0.5, 0.0]] 1 true true false false -0.11111111111111113 0 1.04188007843521944679127450189 ["ModularForm/GL2/Q/holomorphic/162/2/c/a/109/1"] "2-162-9.4-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "9.4" [] [[0.5, 0.0]] 1 true true false false -0.2777777777777778 0 1.24707914124000999913542719082 ["ModularForm/GL2/Q/holomorphic/162/2/c/c/109/1"] "2-162-9.4-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "9.4" [] [[0.5, 0.0]] 1 true true false false -0.11111111111111113 0 1.91882084049818752724148443255 ["ModularForm/GL2/Q/holomorphic/162/2/c/d/109/1"] "2-162-9.4-c1-0-3" 1.1373550513119213 1.293576512744743 2 162 "9.4" [] [[0.5, 0.0]] 1 true true false false 0.2222222222222222 0 2.26057032192534261335558596218 ["ModularForm/GL2/Q/holomorphic/162/2/c/b/109/1"] "2-162-9.7-c1-0-0" 1.1373550513119213 1.293576512744743 2 162 "9.7" [] [[0.5, 0.0]] 1 true true false false -0.2222222222222222 0 1.32677919860022877648496519353 ["ModularForm/GL2/Q/holomorphic/162/2/c/b/55/1"] "2-162-9.7-c1-0-1" 1.1373550513119213 1.293576512744743 2 162 "9.7" [] [[0.5, 0.0]] 1 true true false false 0.11111111111111113 0 2.15987477796209772498727713910 ["ModularForm/GL2/Q/holomorphic/162/2/c/d/55/1"] "2-162-9.7-c1-0-2" 1.1373550513119213 1.293576512744743 2 162 "9.7" [] [[0.5, 0.0]] 1 true true false false 0.11111111111111113 0 2.30857988801074308456157808772 ["ModularForm/GL2/Q/holomorphic/162/2/c/a/55/1"] "2-162-9.7-c1-0-3" 1.1373550513119213 1.293576512744743 2 162 "9.7" [] [[0.5, 0.0]] 1 true true false false 0.2777777777777778 0 2.91754008419131711827962264885 ["ModularForm/GL2/Q/holomorphic/162/2/c/c/55/1"] # 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 $1$. #arith (algebraic) -- # An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers. #$\mathbb{Q}$ (rational) -- # A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes # \[ # L(s)=\prod_pL_p(p^{-s})^{-1} # \] # with $L_p\in \Z[T]$. #self-dual (self_dual) -- # An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real. #$\operatorname{Arg}(\epsilon)$ (root_angle) -- # The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$. #$r$ (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).