# lfunc_search downloaded from the LMFDB on 25 June 2026. # Search link: https://www.lmfdb.org/L/2/9300/5.4 # Query "{'degree': 2, 'conductor': 9300}" returned 186 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. "2-9300-1.1-c1-0-0" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.17089915726945526669273783286 ["ModularForm/GL2/Q/holomorphic/9300/2/a/z/1/2"] "2-9300-1.1-c1-0-1" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.34744625492784160058869810284 ["ModularForm/GL2/Q/holomorphic/9300/2/a/z/1/1"] "2-9300-1.1-c1-0-10" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.55303744041078786108096337861 ["ModularForm/GL2/Q/holomorphic/9300/2/a/x/1/2"] "2-9300-1.1-c1-0-11" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.55998027266185624955809940685 ["ModularForm/GL2/Q/holomorphic/9300/2/a/n/1/2"] "2-9300-1.1-c1-0-12" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.57588768312266425276435589020 ["ModularForm/GL2/Q/holomorphic/9300/2/a/y/1/3"] "2-9300-1.1-c1-0-13" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.57590400178220054847895308641 ["ModularForm/GL2/Q/holomorphic/9300/2/a/ba/1/2"] "2-9300-1.1-c1-0-14" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.57882713905970231461381229355 ["ModularForm/GL2/Q/holomorphic/9300/2/a/t/1/1"] "2-9300-1.1-c1-0-15" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.57987308962593093740528030735 ["ModularForm/GL2/Q/holomorphic/9300/2/a/z/1/4"] "2-9300-1.1-c1-0-16" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.63020075626317806718125011740 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bd/1/7"] "2-9300-1.1-c1-0-17" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.63663707702932697443396600546 ["ModularForm/GL2/Q/holomorphic/9300/2/a/z/1/5"] "2-9300-1.1-c1-0-18" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.63775399652097030718859771609 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bf/1/1"] "2-9300-1.1-c1-0-19" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.64734855310069178694803443150 ["ModularForm/GL2/Q/holomorphic/9300/2/a/y/1/4"] "2-9300-1.1-c1-0-2" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.34922882357655367952665967562 ["ModularForm/GL2/Q/holomorphic/9300/2/a/y/1/5"] "2-9300-1.1-c1-0-20" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.67922070917768276605036691188 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bb/1/3"] "2-9300-1.1-c1-0-21" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.68577731659646158328370026019 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bd/1/1"] "2-9300-1.1-c1-0-22" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.69312049915975333462779714649 ["ModularForm/GL2/Q/holomorphic/9300/2/a/p/1/1"] "2-9300-1.1-c1-0-23" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.70799979387566350948509083005 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bf/1/3"] "2-9300-1.1-c1-0-24" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.72051754409225449210416768779 ["ModularForm/GL2/Q/holomorphic/9300/2/a/t/1/2"] "2-9300-1.1-c1-0-25" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.73166785230508795537288054853 ["ModularForm/GL2/Q/holomorphic/9300/2/a/z/1/6"] "2-9300-1.1-c1-0-26" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.74403157664167519036798737877 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bf/1/4"] "2-9300-1.1-c1-0-27" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.74996097132332904305756788889 ["ModularForm/GL2/Q/holomorphic/9300/2/a/t/1/3"] "2-9300-1.1-c1-0-28" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.75325516685320026180198724316 ["ModularForm/GL2/Q/holomorphic/9300/2/a/ba/1/4"] "2-9300-1.1-c1-0-29" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.75894969862638083452964914873 ["ModularForm/GL2/Q/holomorphic/9300/2/a/z/1/3"] "2-9300-1.1-c1-0-3" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.36321197953492863948900825269 ["ModularForm/GL2/Q/holomorphic/9300/2/a/y/1/2"] "2-9300-1.1-c1-0-30" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.77746202810848414400083475004 ["ModularForm/GL2/Q/holomorphic/9300/2/a/x/1/1"] "2-9300-1.1-c1-0-31" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.795100760696013827644119652083 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bf/1/2"] "2-9300-1.1-c1-0-32" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.805112380110933515728906409191 ["ModularForm/GL2/Q/holomorphic/9300/2/a/x/1/4"] "2-9300-1.1-c1-0-33" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.818486241344852912679539657746 ["ModularForm/GL2/Q/holomorphic/9300/2/a/p/1/2"] "2-9300-1.1-c1-0-34" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.848428346685741714396946806612 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bd/1/6"] "2-9300-1.1-c1-0-35" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.854667038378689526615026605236 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bb/1/5"] "2-9300-1.1-c1-0-36" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.871781064615676025925392817642 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bd/1/5"] "2-9300-1.1-c1-0-37" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.895406682111735126164028840329 ["ModularForm/GL2/Q/holomorphic/9300/2/a/ba/1/6"] "2-9300-1.1-c1-0-38" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.898002964662371027807389617559 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bd/1/4"] "2-9300-1.1-c1-0-39" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.904355668250851425416019684834 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bf/1/7"] "2-9300-1.1-c1-0-4" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.37793358895524919746955368632 ["ModularForm/GL2/Q/holomorphic/9300/2/a/y/1/1"] "2-9300-1.1-c1-0-40" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.905065777732089944088187940502 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bb/1/4"] "2-9300-1.1-c1-0-41" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.908760368480187093678905335030 ["ModularForm/GL2/Q/holomorphic/9300/2/a/y/1/6"] "2-9300-1.1-c1-0-42" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.912214646054641848865846232290 ["ModularForm/GL2/Q/holomorphic/9300/2/a/ba/1/3"] "2-9300-1.1-c1-0-43" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.920285835675193742640019273300 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bb/1/2"] "2-9300-1.1-c1-0-44" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.924846289929323085626118958387 ["EllipticCurve/Q/9300/h", "ModularForm/GL2/Q/holomorphic/9300/2/a/h/1/1", "ModularForm/GL2/Q/holomorphic/9300/2/a/h"] "2-9300-1.1-c1-0-45" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.953789177152747545105849727309 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bc/1/1"] "2-9300-1.1-c1-0-46" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.957584095738522891211868265904 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bf/1/5"] "2-9300-1.1-c1-0-47" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 0.979334475668074714872119260682 ["EllipticCurve/Q/9300/a", "ModularForm/GL2/Q/holomorphic/9300/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/9300/2/a/a"] "2-9300-1.1-c1-0-48" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.00778053719785233372264492470 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bb/1/6"] "2-9300-1.1-c1-0-49" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.02333084201461789930550648612 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bc/1/4"] "2-9300-1.1-c1-0-5" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.40719904314379385638941589312 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bb/1/1"] "2-9300-1.1-c1-0-50" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.02938230697506083335543850570 ["ModularForm/GL2/Q/holomorphic/9300/2/a/s/1/1"] "2-9300-1.1-c1-0-51" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.05568367639555084055773336895 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bc/1/2"] "2-9300-1.1-c1-0-52" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.06956179906425743584593119591 ["EllipticCurve/Q/9300/m", "ModularForm/GL2/Q/holomorphic/9300/2/a/m/1/1", "ModularForm/GL2/Q/holomorphic/9300/2/a/m"] "2-9300-1.1-c1-0-53" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.08238789420473574257822552851 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bf/1/6"] "2-9300-1.1-c1-0-54" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.09079696923655634106944271913 ["ModularForm/GL2/Q/holomorphic/9300/2/a/u/1/2"] "2-9300-1.1-c1-0-55" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.09385669786786343451101122301 ["ModularForm/GL2/Q/holomorphic/9300/2/a/o/1/1"] "2-9300-1.1-c1-0-56" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.11481754364819723296519425117 ["ModularForm/GL2/Q/holomorphic/9300/2/a/ba/1/5"] "2-9300-1.1-c1-0-57" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.13219281094609059981810104247 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bc/1/3"] "2-9300-1.1-c1-0-58" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.13893172478950440276216274202 ["ModularForm/GL2/Q/holomorphic/9300/2/a/x/1/3"] "2-9300-1.1-c1-0-59" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.15926741657025899105808825143 ["ModularForm/GL2/Q/holomorphic/9300/2/a/s/1/2"] "2-9300-1.1-c1-0-6" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.43274042907071848879538566170 ["ModularForm/GL2/Q/holomorphic/9300/2/a/bd/1/2"] "2-9300-1.1-c1-0-60" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.16900765421977176867345974751 ["EllipticCurve/Q/9300/c", "ModularForm/GL2/Q/holomorphic/9300/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/9300/2/a/c"] "2-9300-1.1-c1-0-61" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.18723507337409373225495669628 ["EllipticCurve/Q/9300/d", "ModularForm/GL2/Q/holomorphic/9300/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/9300/2/a/d"] "2-9300-1.1-c1-0-62" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.18856302618762972219282118363 ["ModularForm/GL2/Q/holomorphic/9300/2/a/s/1/3"] "2-9300-1.1-c1-0-63" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.20748722186732770230240332662 ["EllipticCurve/Q/9300/i", "ModularForm/GL2/Q/holomorphic/9300/2/a/i/1/1", "ModularForm/GL2/Q/holomorphic/9300/2/a/i"] "2-9300-1.1-c1-0-64" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.21492151372998824060115266064 ["ModularForm/GL2/Q/holomorphic/9300/2/a/u/1/3"] "2-9300-1.1-c1-0-65" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.22619070632607291405330109558 ["ModularForm/GL2/Q/holomorphic/9300/2/a/u/1/1"] "2-9300-1.1-c1-0-66" 8.617474913209252 74.2608738797908 2 9300 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.22697069374475578879188811685 ["EllipticCurve/Q/9300/e", 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"2-9300-5.4-c1-0-86" 8.617474913209252 74.2608738797908 2 9300 "5.4" [] [[0.5, 0.0]] 1 true true false false -0.42620819117478337 0 1.46513644741409683214287326090 ["ModularForm/GL2/Q/holomorphic/9300/2/g/i/3349/1"] "2-9300-5.4-c1-0-87" 8.617474913209252 74.2608738797908 2 9300 "5.4" [] [[0.5, 0.0]] 1 true true false false -0.42620819117478337 0 1.47374617464409127009211713790 ["ModularForm/GL2/Q/holomorphic/9300/2/g/r/3349/1"] "2-9300-5.4-c1-0-88" 8.617474913209252 74.2608738797908 2 9300 "5.4" [] [[0.5, 0.0]] 1 true true false false -0.32379180882521663 1 1.47994496229530016719143316120 ["ModularForm/GL2/Q/holomorphic/9300/2/g/f/3349/1"] "2-9300-5.4-c1-0-89" 8.617474913209252 74.2608738797908 2 9300 "5.4" [] [[0.5, 0.0]] 1 true true false false 0.42620819117478337 0 1.48884905858165998813987143967 ["ModularForm/GL2/Q/holomorphic/9300/2/g/r/3349/4"] "2-9300-5.4-c1-0-9" 8.617474913209252 74.2608738797908 2 9300 "5.4" [] [[0.5, 0.0]] 1 true true false false -0.17620819117478337 0 0.22759208047836591163682072061 ["ModularForm/GL2/Q/holomorphic/9300/2/g/u/3349/2"] "2-9300-5.4-c1-0-90" 8.617474913209252 74.2608738797908 2 9300 "5.4" [] [[0.5, 0.0]] 1 true true false false -0.42620819117478337 0 1.63915707597101828373255758824 ["ModularForm/GL2/Q/holomorphic/9300/2/g/s/3349/5"] "2-9300-5.4-c1-0-91" 8.617474913209252 74.2608738797908 2 9300 "5.4" [] [[0.5, 0.0]] 1 true true false false -0.32379180882521663 0 1.78273531643524553882084754334 ["ModularForm/GL2/Q/holomorphic/9300/2/g/o/3349/1"] # 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).