# lfunc_search downloaded from the LMFDB on 25 May 2026. # Search link: https://www.lmfdb.org/L/2/5800/1.1/c1-0 # Query "{'degree': 2, 'conductor': 5800, 'spectral_label': 'c1-0'}" returned 133 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. "2-5800-1.1-c1-0-0" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.17879833737638923931607415763 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bd/1/1"] "2-5800-1.1-c1-0-1" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.19695442948122758922100480839 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bc/1/3"] "2-5800-1.1-c1-0-10" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.40284020276094915150427490549 ["ModularForm/GL2/Q/holomorphic/5800/2/a/t/1/1"] "2-5800-1.1-c1-0-100" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.33294990290787368966078313996 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bg/1/6"] "2-5800-1.1-c1-0-101" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.34480502493917287917456278971 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bf/1/7"] "2-5800-1.1-c1-0-102" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.35130757548814899976645980143 ["ModularForm/GL2/Q/holomorphic/5800/2/a/be/1/5"] "2-5800-1.1-c1-0-103" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.35152377999281625744245562322 ["ModularForm/GL2/Q/holomorphic/5800/2/a/o/1/2"] "2-5800-1.1-c1-0-104" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.35791781187305797023248657548 ["ModularForm/GL2/Q/holomorphic/5800/2/a/x/1/4"] "2-5800-1.1-c1-0-105" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.36209553730488331006295885444 ["ModularForm/GL2/Q/holomorphic/5800/2/a/z/1/3"] "2-5800-1.1-c1-0-106" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.39761134498658558083146505487 ["ModularForm/GL2/Q/holomorphic/5800/2/a/y/1/1"] "2-5800-1.1-c1-0-107" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.44895569075883336893363158252 ["ModularForm/GL2/Q/holomorphic/5800/2/a/s/1/2"] "2-5800-1.1-c1-0-108" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.45281031475406199003899505862 ["ModularForm/GL2/Q/holomorphic/5800/2/a/z/1/5"] "2-5800-1.1-c1-0-109" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.46630934825438658559288289812 ["ModularForm/GL2/Q/holomorphic/5800/2/a/q/1/3"] "2-5800-1.1-c1-0-11" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.41813117199594951519734652996 ["ModularForm/GL2/Q/holomorphic/5800/2/a/v/1/3"] "2-5800-1.1-c1-0-110" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.47379741829520604358491933839 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bg/1/7"] "2-5800-1.1-c1-0-111" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.51413594814405459824998878780 ["EllipticCurve/Q/5800/l", "ModularForm/GL2/Q/holomorphic/5800/2/a/l/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/l"] "2-5800-1.1-c1-0-112" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.52798334029882344020429089434 ["ModularForm/GL2/Q/holomorphic/5800/2/a/w/1/5"] "2-5800-1.1-c1-0-113" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.53798935601520779658996725632 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bg/1/10"] "2-5800-1.1-c1-0-114" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.54072915074001985950215384322 ["ModularForm/GL2/Q/holomorphic/5800/2/a/y/1/5"] "2-5800-1.1-c1-0-115" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.56594199940306504094566894399 ["ModularForm/GL2/Q/holomorphic/5800/2/a/be/1/6"] "2-5800-1.1-c1-0-116" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.58067929415160760453576927471 ["ModularForm/GL2/Q/holomorphic/5800/2/a/be/1/7"] "2-5800-1.1-c1-0-117" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.58731104084704386241427582670 ["ModularForm/GL2/Q/holomorphic/5800/2/a/w/1/6"] "2-5800-1.1-c1-0-118" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.59785604331569179989056916906 ["EllipticCurve/Q/5800/m", "ModularForm/GL2/Q/holomorphic/5800/2/a/m/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/m"] "2-5800-1.1-c1-0-119" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.60420280086090731579511054307 ["ModularForm/GL2/Q/holomorphic/5800/2/a/q/1/2"] "2-5800-1.1-c1-0-12" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.42496057968074551971189265994 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bh/1/3"] "2-5800-1.1-c1-0-120" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.62275687453353332072135632145 ["ModularForm/GL2/Q/holomorphic/5800/2/a/be/1/8"] "2-5800-1.1-c1-0-121" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.63240116308749377838632619124 ["ModularForm/GL2/Q/holomorphic/5800/2/a/x/1/5"] "2-5800-1.1-c1-0-122" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.64052033176900355898986058676 ["ModularForm/GL2/Q/holomorphic/5800/2/a/y/1/4"] "2-5800-1.1-c1-0-123" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.66284393361724860146830907267 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bg/1/9"] "2-5800-1.1-c1-0-124" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.67847732970190104608965525501 ["ModularForm/GL2/Q/holomorphic/5800/2/a/u/1/5"] "2-5800-1.1-c1-0-125" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.68728715123410623972058608110 ["ModularForm/GL2/Q/holomorphic/5800/2/a/p/1/3"] "2-5800-1.1-c1-0-126" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.69320603167493268029999909289 ["ModularForm/GL2/Q/holomorphic/5800/2/a/z/1/6"] "2-5800-1.1-c1-0-127" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.72265696342638540603366410696 ["ModularForm/GL2/Q/holomorphic/5800/2/a/z/1/4"] "2-5800-1.1-c1-0-128" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.73580457568191962907999230522 ["ModularForm/GL2/Q/holomorphic/5800/2/a/x/1/6"] "2-5800-1.1-c1-0-129" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.77197246921821185867243572458 ["ModularForm/GL2/Q/holomorphic/5800/2/a/s/1/3"] "2-5800-1.1-c1-0-13" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.48534583968525808465183550801 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bd/1/4"] "2-5800-1.1-c1-0-130" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.79007546526771304430025958412 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bg/1/11"] "2-5800-1.1-c1-0-131" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.84485653295085798611585066480 ["ModularForm/GL2/Q/holomorphic/5800/2/a/y/1/6"] "2-5800-1.1-c1-0-132" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.84705236293524905224973559030 ["ModularForm/GL2/Q/holomorphic/5800/2/a/u/1/4"] "2-5800-1.1-c1-0-14" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.50842700388656861350730957815 ["ModularForm/GL2/Q/holomorphic/5800/2/a/ba/1/4"] "2-5800-1.1-c1-0-15" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.51579947385676897375000844947 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bd/1/2"] "2-5800-1.1-c1-0-16" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.53669098287970835415602206796 ["ModularForm/GL2/Q/holomorphic/5800/2/a/ba/1/5"] "2-5800-1.1-c1-0-17" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.53944618493662514556937772009 ["ModularForm/GL2/Q/holomorphic/5800/2/a/r/1/2"] "2-5800-1.1-c1-0-18" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.56412911448373359252999509863 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bc/1/5"] "2-5800-1.1-c1-0-19" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.56429200832048927905743988392 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bc/1/1"] "2-5800-1.1-c1-0-2" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.20202783763127801603587482664 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bb/1/4"] "2-5800-1.1-c1-0-20" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.58218716913097496110812958759 ["ModularForm/GL2/Q/holomorphic/5800/2/a/v/1/2"] "2-5800-1.1-c1-0-21" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.58408975449964908065751933249 ["ModularForm/GL2/Q/holomorphic/5800/2/a/r/1/1"] "2-5800-1.1-c1-0-22" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.59024449713224229257734947738 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bd/1/5"] "2-5800-1.1-c1-0-23" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.60912750332285794345526906870 ["EllipticCurve/Q/5800/f", "ModularForm/GL2/Q/holomorphic/5800/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/f"] "2-5800-1.1-c1-0-24" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.63837606499017492060184970014 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bf/1/3"] "2-5800-1.1-c1-0-25" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.63970060598711714176299344289 ["EllipticCurve/Q/5800/i", "ModularForm/GL2/Q/holomorphic/5800/2/a/i/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/i"] "2-5800-1.1-c1-0-26" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.64023378801367561288358257558 ["ModularForm/GL2/Q/holomorphic/5800/2/a/o/1/1"] "2-5800-1.1-c1-0-27" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.67350893471971409500577921382 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bc/1/2"] "2-5800-1.1-c1-0-28" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.67484535954876584153402699552 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bh/1/6"] "2-5800-1.1-c1-0-29" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.68182158826672339484009049858 ["ModularForm/GL2/Q/holomorphic/5800/2/a/ba/1/1"] "2-5800-1.1-c1-0-3" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.22143833375352420495591050034 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bh/1/5"] "2-5800-1.1-c1-0-30" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.69387122235846275881547434719 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bb/1/1"] "2-5800-1.1-c1-0-31" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.70380461308625452805900971592 ["EllipticCurve/Q/5800/j", "ModularForm/GL2/Q/holomorphic/5800/2/a/j/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/j"] "2-5800-1.1-c1-0-32" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.70971953735140719766051154555 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bh/1/9"] "2-5800-1.1-c1-0-33" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.71763022703831482014790469108 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bh/1/1"] "2-5800-1.1-c1-0-34" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.71862407184632525700338374947 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bf/1/2"] "2-5800-1.1-c1-0-35" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.74670094866716693026222810075 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bf/1/6"] "2-5800-1.1-c1-0-36" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.75373095013843048140081623188 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bd/1/3"] "2-5800-1.1-c1-0-37" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.75927261610885969979763089229 ["EllipticCurve/Q/5800/c", "ModularForm/GL2/Q/holomorphic/5800/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/c"] "2-5800-1.1-c1-0-38" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.790991324540592738166962833416 ["EllipticCurve/Q/5800/d", "ModularForm/GL2/Q/holomorphic/5800/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/d"] "2-5800-1.1-c1-0-39" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.794411463956646503462063777244 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bb/1/3"] "2-5800-1.1-c1-0-4" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.29536152883002921729474604146 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bb/1/2"] "2-5800-1.1-c1-0-40" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.798565760463143133940961809588 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bf/1/1"] "2-5800-1.1-c1-0-41" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.846908748372614728092707318927 ["ModularForm/GL2/Q/holomorphic/5800/2/a/u/1/1"] "2-5800-1.1-c1-0-42" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.847454348524129830553491365411 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bc/1/4"] "2-5800-1.1-c1-0-43" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.854051952127532214684642575312 ["ModularForm/GL2/Q/holomorphic/5800/2/a/t/1/5"] "2-5800-1.1-c1-0-44" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.858805545832339937165868546053 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bh/1/7"] "2-5800-1.1-c1-0-45" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.860619389229295038276920134105 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bg/1/2"] "2-5800-1.1-c1-0-46" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.860689321933893200676168448760 ["ModularForm/GL2/Q/holomorphic/5800/2/a/t/1/2"] "2-5800-1.1-c1-0-47" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.862659674839283638143350793414 ["ModularForm/GL2/Q/holomorphic/5800/2/a/t/1/4"] "2-5800-1.1-c1-0-48" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.871559529516849836793975615388 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bb/1/5"] "2-5800-1.1-c1-0-49" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.881101747399929115370695190952 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bf/1/5"] "2-5800-1.1-c1-0-5" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.32434116134837925988655758858 ["ModularForm/GL2/Q/holomorphic/5800/2/a/v/1/1"] "2-5800-1.1-c1-0-50" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.902921203476161113754825442152 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bb/1/7"] "2-5800-1.1-c1-0-51" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 0.906127559401400389456999505674 ["EllipticCurve/Q/5800/a", "ModularForm/GL2/Q/holomorphic/5800/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/a"] "2-5800-1.1-c1-0-52" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.911271379393016335662689200185 ["EllipticCurve/Q/5800/k", "ModularForm/GL2/Q/holomorphic/5800/2/a/k/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/k"] "2-5800-1.1-c1-0-53" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.936782620227192863854906354492 ["ModularForm/GL2/Q/holomorphic/5800/2/a/w/1/3"] "2-5800-1.1-c1-0-54" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.950908031491278324781162868601 ["ModularForm/GL2/Q/holomorphic/5800/2/a/ba/1/6"] "2-5800-1.1-c1-0-55" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.952103080224582784412581596347 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bh/1/4"] "2-5800-1.1-c1-0-56" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.959562750024217756573721472721 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bg/1/4"] "2-5800-1.1-c1-0-57" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.965108844212947588541173989424 ["ModularForm/GL2/Q/holomorphic/5800/2/a/v/1/4"] "2-5800-1.1-c1-0-58" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.971416042358064333439503026585 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bh/1/8"] "2-5800-1.1-c1-0-59" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.990627187239678341365909626803 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bc/1/6"] "2-5800-1.1-c1-0-6" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.32775693264435470903698818699 ["ModularForm/GL2/Q/holomorphic/5800/2/a/t/1/3"] "2-5800-1.1-c1-0-60" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.997031954718535147788492990452 ["EllipticCurve/Q/5800/n", "ModularForm/GL2/Q/holomorphic/5800/2/a/n/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/n"] "2-5800-1.1-c1-0-61" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.998757068668786211451656889490 ["ModularForm/GL2/Q/holomorphic/5800/2/a/x/1/3"] "2-5800-1.1-c1-0-62" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.00780630241412522867428275592 ["ModularForm/GL2/Q/holomorphic/5800/2/a/be/1/2"] "2-5800-1.1-c1-0-63" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.01249478601454709602837478061 ["ModularForm/GL2/Q/holomorphic/5800/2/a/be/1/1"] "2-5800-1.1-c1-0-64" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.05897708034417991770495944781 ["ModularForm/GL2/Q/holomorphic/5800/2/a/z/1/1"] "2-5800-1.1-c1-0-65" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.06017737021434199453814467594 ["ModularForm/GL2/Q/holomorphic/5800/2/a/r/1/3"] "2-5800-1.1-c1-0-66" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.06692114427373365573003839911 ["EllipticCurve/Q/5800/b", "ModularForm/GL2/Q/holomorphic/5800/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/b"] "2-5800-1.1-c1-0-67" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.07368787057639147268586645881 ["ModularForm/GL2/Q/holomorphic/5800/2/a/ba/1/3"] "2-5800-1.1-c1-0-68" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.07744490522651184657775412137 ["ModularForm/GL2/Q/holomorphic/5800/2/a/x/1/1"] "2-5800-1.1-c1-0-69" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.09026589458373742317618002932 ["ModularForm/GL2/Q/holomorphic/5800/2/a/be/1/4"] "2-5800-1.1-c1-0-7" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.36896130036580730607589453657 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bh/1/2"] "2-5800-1.1-c1-0-70" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.09578973362138581108129531281 ["ModularForm/GL2/Q/holomorphic/5800/2/a/p/1/2"] "2-5800-1.1-c1-0-71" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.10130738368877728327654530613 ["ModularForm/GL2/Q/holomorphic/5800/2/a/y/1/2"] "2-5800-1.1-c1-0-72" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.10596454577136408594700761267 ["ModularForm/GL2/Q/holomorphic/5800/2/a/u/1/2"] "2-5800-1.1-c1-0-73" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.11423180316331737597606965478 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bd/1/6"] "2-5800-1.1-c1-0-74" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.11576061950321800636197346973 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bf/1/8"] "2-5800-1.1-c1-0-75" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.11650776132960233265609010863 ["ModularForm/GL2/Q/holomorphic/5800/2/a/p/1/1"] "2-5800-1.1-c1-0-76" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.12337965961401808064257912895 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bd/1/7"] "2-5800-1.1-c1-0-77" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.13868235102052499257336944609 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bb/1/6"] "2-5800-1.1-c1-0-78" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.15017898132974381259994889582 ["EllipticCurve/Q/5800/e", "ModularForm/GL2/Q/holomorphic/5800/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/e"] "2-5800-1.1-c1-0-79" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.15265209259589906461216606975 ["ModularForm/GL2/Q/holomorphic/5800/2/a/w/1/1"] "2-5800-1.1-c1-0-8" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.37257054694040329563863518464 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bf/1/4"] "2-5800-1.1-c1-0-80" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.17137720574548429521624151998 ["ModularForm/GL2/Q/holomorphic/5800/2/a/v/1/5"] "2-5800-1.1-c1-0-81" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.17743891945698969759170048528 ["ModularForm/GL2/Q/holomorphic/5800/2/a/q/1/1"] "2-5800-1.1-c1-0-82" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.17916042251712065296926851044 ["ModularForm/GL2/Q/holomorphic/5800/2/a/x/1/2"] "2-5800-1.1-c1-0-83" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.21427096220638702343168152343 ["ModularForm/GL2/Q/holomorphic/5800/2/a/y/1/3"] "2-5800-1.1-c1-0-84" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.22206946745406320344687055763 ["ModularForm/GL2/Q/holomorphic/5800/2/a/be/1/3"] "2-5800-1.1-c1-0-85" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.22709179673157792268385735183 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bg/1/8"] "2-5800-1.1-c1-0-86" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.22829085317134420471529697923 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bg/1/3"] "2-5800-1.1-c1-0-87" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.22881362350292948229071799111 ["ModularForm/GL2/Q/holomorphic/5800/2/a/s/1/1"] "2-5800-1.1-c1-0-88" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.23021989904350641214386299022 ["EllipticCurve/Q/5800/g", "ModularForm/GL2/Q/holomorphic/5800/2/a/g/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/g"] "2-5800-1.1-c1-0-89" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.23135772465920530982683477015 ["ModularForm/GL2/Q/holomorphic/5800/2/a/z/1/2"] "2-5800-1.1-c1-0-9" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.40035807308148686348437281086 ["ModularForm/GL2/Q/holomorphic/5800/2/a/ba/1/2"] "2-5800-1.1-c1-0-90" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.23423020514614932308572389785 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bg/1/1"] "2-5800-1.1-c1-0-91" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.23569498027715685279699177696 ["ModularForm/GL2/Q/holomorphic/5800/2/a/ba/1/7"] "2-5800-1.1-c1-0-92" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.23815673702447026014450693438 ["ModularForm/GL2/Q/holomorphic/5800/2/a/w/1/2"] "2-5800-1.1-c1-0-93" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.23894024937379849395264076064 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bh/1/11"] "2-5800-1.1-c1-0-94" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.25821567336773206157596349138 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bc/1/7"] "2-5800-1.1-c1-0-95" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.27927406163109211552586491787 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bh/1/10"] "2-5800-1.1-c1-0-96" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.29858560962261668955369720569 ["ModularForm/GL2/Q/holomorphic/5800/2/a/bg/1/5"] "2-5800-1.1-c1-0-97" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.30917815158706569005591571881 ["ModularForm/GL2/Q/holomorphic/5800/2/a/w/1/4"] "2-5800-1.1-c1-0-98" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.31195145109566964956390879900 ["ModularForm/GL2/Q/holomorphic/5800/2/a/u/1/3"] "2-5800-1.1-c1-0-99" 6.8053826617129065 46.31323317234264 2 5800 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.31884252077796023074643546983 ["EllipticCurve/Q/5800/h", "ModularForm/GL2/Q/holomorphic/5800/2/a/h/1/1", "ModularForm/GL2/Q/holomorphic/5800/2/a/h"] # 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).