# lfunc_search downloaded from the LMFDB on 26 August 2026. # Search link: https://www.lmfdb.org/L/rational/4/1008^2 # Query "{'degree': 4, 'conductor': 1016064, 'rational': True}" returned 216 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. "4-1008e2-1.1-c0e2-0-0" 0.7092654881876507 0.2530668812404864 4 1016064 "1.1" [] [[0.0, 0.0], [0.0, 0.0]] 0 false 1 0 0.986056299792670216456693117002 ["ModularForm/GL2/Q/holomorphic/1008/1/cg/a"] "4-1008e2-1.1-c0e2-0-1" 0.7092654881876507 0.2530668812404864 4 1016064 "1.1" [] [[0.0, 0.0], [0.0, 0.0]] 0 false 1 0 1.05803658760874879022386712685 ["ModularForm/GL2/Q/holomorphic/1008/1/u/b"] "4-1008e2-1.1-c0e2-0-2" 0.7092654881876507 0.2530668812404864 4 1016064 "1.1" [] [[0.0, 0.0], [0.0, 0.0]] 0 false 1 0 1.12320597885965016683311254231 ["ModularForm/GL2/Q/holomorphic/1008/1/u/a"] "4-1008e2-1.1-c0e2-0-3" 0.7092654881876507 0.2530668812404864 4 1016064 "1.1" [] [[0.0, 0.0], [0.0, 0.0]] 0 false 1 0 1.12867824155953033217823816685 ["ModularForm/GL2/Q/holomorphic/1008/1/cd/a"] "4-1008e2-1.1-c0e2-0-4" 0.7092654881876507 0.2530668812404864 4 1016064 "1.1" [] [[0.0, 0.0], [0.0, 0.0]] 0 false 1 0 1.24909024554417592940450729484 ["ModularForm/GL2/Q/holomorphic/1008/1/cd/b"] "4-1008e2-1.1-c0e2-0-5" 0.7092654881876507 0.2530668812404864 4 1016064 "1.1" [] [[0.0, 0.0], [0.0, 0.0]] 0 false 1 0 1.57704092169740242446897586579 ["ModularForm/GL2/Q/holomorphic/1008/1/u/c"] "4-1008e2-1.1-c1e2-0-0" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.085466292218050886097484326081 ["ModularForm/GL2/Q/holomorphic/1008/2/cx/f"] "4-1008e2-1.1-c1e2-0-1" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.19068555869781792360466605670 ["ModularForm/GL2/Q/holomorphic/1008/2/cs/e"] "4-1008e2-1.1-c1e2-0-10" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 0.35818536227499969622181162380971 ["EllipticCurve/2.0.7.1/20736.6/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.4/a", "EllipticCurve/2.0.7.1/20736.4/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.6/a"] "4-1008e2-1.1-c1e2-0-100" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.05807239814880289810880782296 ["ModularForm/GL2/Q/holomorphic/1008/2/cz/c"] "4-1008e2-1.1-c1e2-0-101" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.06071389403996790408828165924 ["ModularForm/GL2/Q/holomorphic/1008/2/s/h"] "4-1008e2-1.1-c1e2-0-102" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.07616601427985978452723732003822 ["EllipticCurve/2.0.7.1/20736.5/a", "EllipticCurve/2.0.7.1/20736.5/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.5/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.5/b"] "4-1008e2-1.1-c1e2-0-103" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.10121837595018112655442827978 ["ModularForm/GL2/Q/holomorphic/1008/2/s/l"] "4-1008e2-1.1-c1e2-0-104" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.10634626536378818837645573291 ["ModularForm/GL2/Q/holomorphic/1008/2/cx/h"] "4-1008e2-1.1-c1e2-0-105" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.11691662754238754333523094655947 ["EllipticCurve/2.0.3.1/112896.1/j", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.3/g", "EllipticCurve/2.0.3.1/112896.3/g", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.1/j"] "4-1008e2-1.1-c1e2-0-106" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.11782706441028832419954424002719 ["EllipticCurve/2.0.3.1/112896.1/m", "EllipticCurve/2.0.3.1/112896.3/m", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.3/m", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.1/m"] "4-1008e2-1.1-c1e2-0-107" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.12488811452032398675148182894495 ["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/v", "EllipticCurve/2.0.3.1/112896.2/v", "EllipticCurve/2.0.3.1/112896.2/n", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/n"] "4-1008e2-1.1-c1e2-0-108" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.12644399783657185408249656998333 ["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/15876.3/a", "EllipticCurve/2.0.8.1/15876.3/a"] "4-1008e2-1.1-c1e2-0-109" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.13371683359032634823422648113 ["ModularForm/GL2/Q/holomorphic/1008/2/q/d"] "4-1008e2-1.1-c1e2-0-11" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.37408800520177173590651995728 ["ModularForm/GL2/Q/holomorphic/1008/2/b/b"] "4-1008e2-1.1-c1e2-0-110" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.13997227720930482129043401051112 ["EllipticCurve/2.0.3.1/112896.1/p", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.1/p", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.3/p", "EllipticCurve/2.0.3.1/112896.3/p"] "4-1008e2-1.1-c1e2-0-111" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.14005858652858160450073611048817 ["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/x", "EllipticCurve/2.0.3.1/112896.2/o", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/o", "EllipticCurve/2.0.3.1/112896.2/x"] "4-1008e2-1.1-c1e2-0-112" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true 1 0 1.14426658727060053008944690862091 ["ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.2/d", "EllipticCurve/2.0.7.1/20736.2/d", "EllipticCurve/2.0.7.1/20736.8/d", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.8/d"] "4-1008e2-1.1-c1e2-0-113" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.14495436909313277848926104846723 ["ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.2/a", "EllipticCurve/2.0.7.1/20736.2/a", "EllipticCurve/2.0.7.1/20736.8/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.8/a"] "4-1008e2-1.1-c1e2-0-114" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.14834259446991635774405137668059 ["EllipticCurve/2.0.3.1/112896.2/c", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/j", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/c", "EllipticCurve/2.0.3.1/112896.2/j"] "4-1008e2-1.1-c1e2-0-115" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.14856923157137855803176925211364 ["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.1/s", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.3/s", "EllipticCurve/2.0.3.1/112896.3/s", "EllipticCurve/2.0.3.1/112896.1/s"] "4-1008e2-1.1-c1e2-0-116" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.15838095726178298934837199777019 ["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.1/h", "EllipticCurve/2.0.3.1/112896.1/h", "EllipticCurve/2.0.3.1/112896.3/l", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.3/l"] "4-1008e2-1.1-c1e2-0-117" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.19305346682000134939625159045533 ["EllipticCurve/2.0.3.1/112896.2/bd", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/bc", "EllipticCurve/2.0.3.1/112896.2/bc", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/bd"] "4-1008e2-1.1-c1e2-0-118" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.22306959456710338102849832242 ["ModularForm/GL2/Q/holomorphic/1008/2/s/p"] "4-1008e2-1.1-c1e2-0-119" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 2 1.24747541155110376980734851879 ["ModularForm/GL2/Q/holomorphic/1008/2/bf/a"] "4-1008e2-1.1-c1e2-0-12" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.38458328176277377067320820992 ["EllipticCurve/2.0.3.1/112896.1/CMg", "EllipticCurve/2.0.3.1/112896.3/CMg", "ModularForm/GL2/Q/holomorphic/1008/2/cs/g"] "4-1008e2-1.1-c1e2-0-120" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.24848984411413032704653694327 ["ModularForm/GL2/Q/holomorphic/1008/2/cs/n"] "4-1008e2-1.1-c1e2-0-121" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false -1 1 1.24950065754749508798543005172354 ["EllipticCurve/2.0.7.1/20736.5/c", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.5/c"] "4-1008e2-1.1-c1e2-0-122" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.26689004357259421401987641260433 ["EllipticCurve/2.0.7.1/20736.6/c", "EllipticCurve/2.0.7.1/20736.4/c", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.4/c", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.6/c"] "4-1008e2-1.1-c1e2-0-123" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.27322242348485215500748441505771 ["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.1/u", "EllipticCurve/2.0.3.1/112896.1/u", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.3/t", "EllipticCurve/2.0.3.1/112896.3/t"] "4-1008e2-1.1-c1e2-0-124" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.27612197911727362815097229605121 ["EllipticCurve/2.0.7.1/20736.3/a", "EllipticCurve/2.0.7.1/20736.7/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.7/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/20736.3/a"] "4-1008e2-1.1-c1e2-0-125" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.27869745249649069868436404365293 ["EllipticCurve/2.0.3.1/112896.1/n", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.1/n", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.3/n", "EllipticCurve/2.0.3.1/112896.3/n"] "4-1008e2-1.1-c1e2-0-126" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 1.28066535480041276747489914259 ["ModularForm/GL2/Q/holomorphic/1008/2/t/f"] "4-1008e2-1.1-c1e2-0-127" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.30166760080318167129992096711508 ["EllipticCurve/2.0.3.1/112896.1/y", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.3/ba", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.1/y", "EllipticCurve/2.0.3.1/112896.3/ba"] "4-1008e2-1.1-c1e2-0-128" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.30807555830537656813855642210718 ["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/p", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/z", "EllipticCurve/2.0.3.1/112896.2/p", "EllipticCurve/2.0.3.1/112896.2/z"] "4-1008e2-1.1-c1e2-0-129" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.30847962068480412570270467754224 ["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.3/q", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.1/q", "EllipticCurve/2.0.3.1/112896.1/q", "EllipticCurve/2.0.3.1/112896.3/q"] "4-1008e2-1.1-c1e2-0-13" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 false 1 0 0.40741115000282173445084097516 ["ModularForm/GL2/Q/holomorphic/1008/2/cj/b"] "4-1008e2-1.1-c1e2-0-130" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.31641358715670263887465079532189 ["EllipticCurve/2.0.3.1/112896.2/bf", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/bf", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/112896.2/be", "EllipticCurve/2.0.3.1/112896.2/be"] "4-1008e2-1.1-c1e2-0-131" 2.8370619527506036 64.78512159756457 4 1016064 "1.1" [] [[0.5, 0.0], [0.5, 0.0]] 1 true -1 1 1.32003037448486622044460642906214 ["EllipticCurve/2.0.3.1/112896.3/z", 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1.18047200305605481156027163761 ["ModularForm/GL2/Q/holomorphic/1008/6/a/bw"] "4-1008e2-1.1-c5e2-0-3" 12.714829545514345 26136.183442385598 4 1016064 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.38858703701340483324043381370 ["ModularForm/GL2/Q/holomorphic/1008/6/a/bu"] "4-1008e2-1.1-c5e2-0-4" 12.714829545514345 26136.183442385598 4 1016064 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.39829986139368402017231136434 ["ModularForm/GL2/Q/holomorphic/1008/6/b/b"] "4-1008e2-1.1-c5e2-0-5" 12.714829545514345 26136.183442385598 4 1016064 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.39933713685501615991879351181 ["ModularForm/GL2/Q/holomorphic/1008/6/b/a"] "4-1008e2-1.1-c5e2-0-6" 12.714829545514345 26136.183442385598 4 1016064 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.44781218490007675307819611395 ["ModularForm/GL2/Q/holomorphic/1008/6/a/bs"] "4-1008e2-1.1-c5e2-0-7" 12.714829545514345 26136.183442385598 4 1016064 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.46722074126278856589504561030 ["ModularForm/GL2/Q/holomorphic/1008/6/a/bp"] "4-1008e2-1.1-c5e2-0-8" 12.714829545514345 26136.183442385598 4 1016064 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.60129495428075790895057899327 ["ModularForm/GL2/Q/holomorphic/1008/6/a/br"] "4-1008e2-1.1-c5e2-0-9" 12.714829545514345 26136.183442385598 4 1016064 "1.1" [] [[2.5, 0.0], [2.5, 0.0]] 5 false 1 0 0.60912811474838416043248964761 ["ModularForm/GL2/Q/holomorphic/1008/6/a/bh"] # 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).