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Properties

Label 24-936e12-1.1-c1e12-0-1
Degree 2424
Conductor 4.522×10354.522\times 10^{35}
Sign 11
Analytic cond. 3.03833×10103.03833\times 10^{10}
Root an. cond. 2.733862.73386
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 5-s + 2·7-s + 9·9-s − 11-s − 6·13-s + 4·15-s + 12·17-s − 14·19-s − 8·21-s + 19·23-s + 18·25-s − 17·27-s − 2·29-s + 8·31-s + 4·33-s − 2·35-s − 34·37-s + 24·39-s − 2·41-s + 15·43-s − 9·45-s − 9·47-s + 29·49-s − 48·51-s + 32·53-s + 55-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.447·5-s + 0.755·7-s + 3·9-s − 0.301·11-s − 1.66·13-s + 1.03·15-s + 2.91·17-s − 3.21·19-s − 1.74·21-s + 3.96·23-s + 18/5·25-s − 3.27·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s − 0.338·35-s − 5.58·37-s + 3.84·39-s − 0.312·41-s + 2.28·43-s − 1.34·45-s − 1.31·47-s + 29/7·49-s − 6.72·51-s + 4.39·53-s + 0.134·55-s + ⋯

Functional equation

Λ(s)=((236⋅324⋅1312)s/2 ΓC(s)12 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((236⋅324⋅1312)s/2 ΓC(s+1/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 236⋅324⋅13122^{36} \cdot 3^{24} \cdot 13^{12}
Sign: 11
Analytic conductor: 3.03833×10103.03833\times 10^{10}
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 236⋅324⋅1312, ( :[1/2]12), 1)(24,\ 2^{36} \cdot 3^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.7417743871.741774387
L(12)L(\frac12) ≈\approx 1.7417743871.741774387
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+4T+7T2+p2T3−pT4−5p2T5−11p2T6−5p3T7−p3T8+p5T9+7p4T10+4p5T11+p6T12 1 + 4 T + 7 T^{2} + p^{2} T^{3} - p T^{4} - 5 p^{2} T^{5} - 11 p^{2} T^{6} - 5 p^{3} T^{7} - p^{3} T^{8} + p^{5} T^{9} + 7 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
13 (1+T+T2)6 ( 1 + T + T^{2} )^{6}
good5 1+T−17T2−6pT3+144T4+356T5−664T6−534pT7+896T8+2532pT9+12814T10−27063T11−103651T12−27063pT13+12814p2T14+2532p4T15+896p4T16−534p6T17−664p6T18+356p7T19+144p8T20−6p10T21−17p10T22+p11T23+p12T24 1 + T - 17 T^{2} - 6 p T^{3} + 144 T^{4} + 356 T^{5} - 664 T^{6} - 534 p T^{7} + 896 T^{8} + 2532 p T^{9} + 12814 T^{10} - 27063 T^{11} - 103651 T^{12} - 27063 p T^{13} + 12814 p^{2} T^{14} + 2532 p^{4} T^{15} + 896 p^{4} T^{16} - 534 p^{6} T^{17} - 664 p^{6} T^{18} + 356 p^{7} T^{19} + 144 p^{8} T^{20} - 6 p^{10} T^{21} - 17 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24}
7 1−2T−25T2+12pT3+303T4−1467T5−1663T6+15712T7−3322T8−104066T9+22695pT10+305685T11−1527553T12+305685pT13+22695p3T14−104066p3T15−3322p4T16+15712p5T17−1663p6T18−1467p7T19+303p8T20+12p10T21−25p10T22−2p11T23+p12T24 1 - 2 T - 25 T^{2} + 12 p T^{3} + 303 T^{4} - 1467 T^{5} - 1663 T^{6} + 15712 T^{7} - 3322 T^{8} - 104066 T^{9} + 22695 p T^{10} + 305685 T^{11} - 1527553 T^{12} + 305685 p T^{13} + 22695 p^{3} T^{14} - 104066 p^{3} T^{15} - 3322 p^{4} T^{16} + 15712 p^{5} T^{17} - 1663 p^{6} T^{18} - 1467 p^{7} T^{19} + 303 p^{8} T^{20} + 12 p^{10} T^{21} - 25 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24}
11 1+T−28T2−145T3+356T4+3846T5+2192T6−60825T7−147607T8+615179T9+3066540T10−2917334T11−40184823T12−2917334pT13+3066540p2T14+615179p3T15−147607p4T16−60825p5T17+2192p6T18+3846p7T19+356p8T20−145p9T21−28p10T22+p11T23+p12T24 1 + T - 28 T^{2} - 145 T^{3} + 356 T^{4} + 3846 T^{5} + 2192 T^{6} - 60825 T^{7} - 147607 T^{8} + 615179 T^{9} + 3066540 T^{10} - 2917334 T^{11} - 40184823 T^{12} - 2917334 p T^{13} + 3066540 p^{2} T^{14} + 615179 p^{3} T^{15} - 147607 p^{4} T^{16} - 60825 p^{5} T^{17} + 2192 p^{6} T^{18} + 3846 p^{7} T^{19} + 356 p^{8} T^{20} - 145 p^{9} T^{21} - 28 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24}
17 (1−6T+72T2−235T3+1812T4−3435T5+30155T6−3435pT7+1812p2T8−235p3T9+72p4T10−6p5T11+p6T12)2 ( 1 - 6 T + 72 T^{2} - 235 T^{3} + 1812 T^{4} - 3435 T^{5} + 30155 T^{6} - 3435 p T^{7} + 1812 p^{2} T^{8} - 235 p^{3} T^{9} + 72 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2}
19 (1+7T+109T2+552T3+4881T4+1000pT5+120760T6+1000p2T7+4881p2T8+552p3T9+109p4T10+7p5T11+p6T12)2 ( 1 + 7 T + 109 T^{2} + 552 T^{3} + 4881 T^{4} + 1000 p T^{5} + 120760 T^{6} + 1000 p^{2} T^{7} + 4881 p^{2} T^{8} + 552 p^{3} T^{9} + 109 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2}
23 1−19T+85T2+76T3+4236T4−54722T5+61228T6+217791T7+9916640T8−48864976T9−72985812T10−431640815T11+8784786536T12−431640815pT13−72985812p2T14−48864976p3T15+9916640p4T16+217791p5T17+61228p6T18−54722p7T19+4236p8T20+76p9T21+85p10T22−19p11T23+p12T24 1 - 19 T + 85 T^{2} + 76 T^{3} + 4236 T^{4} - 54722 T^{5} + 61228 T^{6} + 217791 T^{7} + 9916640 T^{8} - 48864976 T^{9} - 72985812 T^{10} - 431640815 T^{11} + 8784786536 T^{12} - 431640815 p T^{13} - 72985812 p^{2} T^{14} - 48864976 p^{3} T^{15} + 9916640 p^{4} T^{16} + 217791 p^{5} T^{17} + 61228 p^{6} T^{18} - 54722 p^{7} T^{19} + 4236 p^{8} T^{20} + 76 p^{9} T^{21} + 85 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24}
29 1+2T−71T2−50T3+2034T4−2945T5−38099T6+174165T7+729554T8−3060184T9+10301124T10+14837524T11−1050511345T12+14837524pT13+10301124p2T14−3060184p3T15+729554p4T16+174165p5T17−38099p6T18−2945p7T19+2034p8T20−50p9T21−71p10T22+2p11T23+p12T24 1 + 2 T - 71 T^{2} - 50 T^{3} + 2034 T^{4} - 2945 T^{5} - 38099 T^{6} + 174165 T^{7} + 729554 T^{8} - 3060184 T^{9} + 10301124 T^{10} + 14837524 T^{11} - 1050511345 T^{12} + 14837524 p T^{13} + 10301124 p^{2} T^{14} - 3060184 p^{3} T^{15} + 729554 p^{4} T^{16} + 174165 p^{5} T^{17} - 38099 p^{6} T^{18} - 2945 p^{7} T^{19} + 2034 p^{8} T^{20} - 50 p^{9} T^{21} - 71 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24}
31 1−8T−77T2+468T3+5264T4−541pT5−212981T6+65633T7+6780536T8+8681576T9−150146070T10−219740916T11+3926987469T12−219740916pT13−150146070p2T14+8681576p3T15+6780536p4T16+65633p5T17−212981p6T18−541p8T19+5264p8T20+468p9T21−77p10T22−8p11T23+p12T24 1 - 8 T - 77 T^{2} + 468 T^{3} + 5264 T^{4} - 541 p T^{5} - 212981 T^{6} + 65633 T^{7} + 6780536 T^{8} + 8681576 T^{9} - 150146070 T^{10} - 219740916 T^{11} + 3926987469 T^{12} - 219740916 p T^{13} - 150146070 p^{2} T^{14} + 8681576 p^{3} T^{15} + 6780536 p^{4} T^{16} + 65633 p^{5} T^{17} - 212981 p^{6} T^{18} - 541 p^{8} T^{19} + 5264 p^{8} T^{20} + 468 p^{9} T^{21} - 77 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24}
37 (1+17T+155T2+747T3+1666T4−4493T5−49898T6−4493pT7+1666p2T8+747p3T9+155p4T10+17p5T11+p6T12)2 ( 1 + 17 T + 155 T^{2} + 747 T^{3} + 1666 T^{4} - 4493 T^{5} - 49898 T^{6} - 4493 p T^{7} + 1666 p^{2} T^{8} + 747 p^{3} T^{9} + 155 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} )^{2}
41 1+2T−37T2+52T3−1099T4−14289T5+36959T6+342990T7−546304T8−12245894T9+83979345T10+483815711T11−4001736225T12+483815711pT13+83979345p2T14−12245894p3T15−546304p4T16+342990p5T17+36959p6T18−14289p7T19−1099p8T20+52p9T21−37p10T22+2p11T23+p12T24 1 + 2 T - 37 T^{2} + 52 T^{3} - 1099 T^{4} - 14289 T^{5} + 36959 T^{6} + 342990 T^{7} - 546304 T^{8} - 12245894 T^{9} + 83979345 T^{10} + 483815711 T^{11} - 4001736225 T^{12} + 483815711 p T^{13} + 83979345 p^{2} T^{14} - 12245894 p^{3} T^{15} - 546304 p^{4} T^{16} + 342990 p^{5} T^{17} + 36959 p^{6} T^{18} - 14289 p^{7} T^{19} - 1099 p^{8} T^{20} + 52 p^{9} T^{21} - 37 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24}
43 1−15T−44T2+1605T3+436T4−96150T5+13421T6+2798805T7+15939163T8−37455615T9−1767812744T10−358014495T11+104035237766T12−358014495pT13−1767812744p2T14−37455615p3T15+15939163p4T16+2798805p5T17+13421p6T18−96150p7T19+436p8T20+1605p9T21−44p10T22−15p11T23+p12T24 1 - 15 T - 44 T^{2} + 1605 T^{3} + 436 T^{4} - 96150 T^{5} + 13421 T^{6} + 2798805 T^{7} + 15939163 T^{8} - 37455615 T^{9} - 1767812744 T^{10} - 358014495 T^{11} + 104035237766 T^{12} - 358014495 p T^{13} - 1767812744 p^{2} T^{14} - 37455615 p^{3} T^{15} + 15939163 p^{4} T^{16} + 2798805 p^{5} T^{17} + 13421 p^{6} T^{18} - 96150 p^{7} T^{19} + 436 p^{8} T^{20} + 1605 p^{9} T^{21} - 44 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24}
47 1+9T−96T2−791T3+8358T4+63018T5−231163T6−2385648T7−5455962T8+94088395T9+1640985885T10−432734379T11−92112254279T12−432734379pT13+1640985885p2T14+94088395p3T15−5455962p4T16−2385648p5T17−231163p6T18+63018p7T19+8358p8T20−791p9T21−96p10T22+9p11T23+p12T24 1 + 9 T - 96 T^{2} - 791 T^{3} + 8358 T^{4} + 63018 T^{5} - 231163 T^{6} - 2385648 T^{7} - 5455962 T^{8} + 94088395 T^{9} + 1640985885 T^{10} - 432734379 T^{11} - 92112254279 T^{12} - 432734379 p T^{13} + 1640985885 p^{2} T^{14} + 94088395 p^{3} T^{15} - 5455962 p^{4} T^{16} - 2385648 p^{5} T^{17} - 231163 p^{6} T^{18} + 63018 p^{7} T^{19} + 8358 p^{8} T^{20} - 791 p^{9} T^{21} - 96 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24}
53 (1−16T+341T2−3597T3+44717T4−351571T5+3140075T6−351571pT7+44717p2T8−3597p3T9+341p4T10−16p5T11+p6T12)2 ( 1 - 16 T + 341 T^{2} - 3597 T^{3} + 44717 T^{4} - 351571 T^{5} + 3140075 T^{6} - 351571 p T^{7} + 44717 p^{2} T^{8} - 3597 p^{3} T^{9} + 341 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2}
59 1−5T−219T2+2256T3+24682T4−382892T5−866756T6+40107018T7−98677820T8−2428428608T9+18619322966T10+64533742905T11−1426618140149T12+64533742905pT13+18619322966p2T14−2428428608p3T15−98677820p4T16+40107018p5T17−866756p6T18−382892p7T19+24682p8T20+2256p9T21−219p10T22−5p11T23+p12T24 1 - 5 T - 219 T^{2} + 2256 T^{3} + 24682 T^{4} - 382892 T^{5} - 866756 T^{6} + 40107018 T^{7} - 98677820 T^{8} - 2428428608 T^{9} + 18619322966 T^{10} + 64533742905 T^{11} - 1426618140149 T^{12} + 64533742905 p T^{13} + 18619322966 p^{2} T^{14} - 2428428608 p^{3} T^{15} - 98677820 p^{4} T^{16} + 40107018 p^{5} T^{17} - 866756 p^{6} T^{18} - 382892 p^{7} T^{19} + 24682 p^{8} T^{20} + 2256 p^{9} T^{21} - 219 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24}
61 1−6T−158T2−452T3+25558T4+100947T5−1054744T6−21449619T7−190172T8+1201558268T9+12652888861T10−51952082111T11−866504801022T12−51952082111pT13+12652888861p2T14+1201558268p3T15−190172p4T16−21449619p5T17−1054744p6T18+100947p7T19+25558p8T20−452p9T21−158p10T22−6p11T23+p12T24 1 - 6 T - 158 T^{2} - 452 T^{3} + 25558 T^{4} + 100947 T^{5} - 1054744 T^{6} - 21449619 T^{7} - 190172 T^{8} + 1201558268 T^{9} + 12652888861 T^{10} - 51952082111 T^{11} - 866504801022 T^{12} - 51952082111 p T^{13} + 12652888861 p^{2} T^{14} + 1201558268 p^{3} T^{15} - 190172 p^{4} T^{16} - 21449619 p^{5} T^{17} - 1054744 p^{6} T^{18} + 100947 p^{7} T^{19} + 25558 p^{8} T^{20} - 452 p^{9} T^{21} - 158 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24}
67 1−8T−87T2+2086T3−7511T4−104423T5+2025145T6−9453174T7−79890902T8+1676113420T9−6418620129T10−56233364259T11+977846727627T12−56233364259pT13−6418620129p2T14+1676113420p3T15−79890902p4T16−9453174p5T17+2025145p6T18−104423p7T19−7511p8T20+2086p9T21−87p10T22−8p11T23+p12T24 1 - 8 T - 87 T^{2} + 2086 T^{3} - 7511 T^{4} - 104423 T^{5} + 2025145 T^{6} - 9453174 T^{7} - 79890902 T^{8} + 1676113420 T^{9} - 6418620129 T^{10} - 56233364259 T^{11} + 977846727627 T^{12} - 56233364259 p T^{13} - 6418620129 p^{2} T^{14} + 1676113420 p^{3} T^{15} - 79890902 p^{4} T^{16} - 9453174 p^{5} T^{17} + 2025145 p^{6} T^{18} - 104423 p^{7} T^{19} - 7511 p^{8} T^{20} + 2086 p^{9} T^{21} - 87 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24}
71 (1+21T+552T2+7365T3+108168T4+1029909T5+10503502T6+1029909pT7+108168p2T8+7365p3T9+552p4T10+21p5T11+p6T12)2 ( 1 + 21 T + 552 T^{2} + 7365 T^{3} + 108168 T^{4} + 1029909 T^{5} + 10503502 T^{6} + 1029909 p T^{7} + 108168 p^{2} T^{8} + 7365 p^{3} T^{9} + 552 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} )^{2}
73 (1+T+181T2+797T3+15046T4+126001T5+1041188T6+126001pT7+15046p2T8+797p3T9+181p4T10+p5T11+p6T12)2 ( 1 + T + 181 T^{2} + 797 T^{3} + 15046 T^{4} + 126001 T^{5} + 1041188 T^{6} + 126001 p T^{7} + 15046 p^{2} T^{8} + 797 p^{3} T^{9} + 181 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2}
79 1−34T+274T2+1008T3+24830T4−899081T5+3408808T6+17123755T7+654386474T8−7963303748T9−4308732765T10−202867350615T11+6976154776464T12−202867350615pT13−4308732765p2T14−7963303748p3T15+654386474p4T16+17123755p5T17+3408808p6T18−899081p7T19+24830p8T20+1008p9T21+274p10T22−34p11T23+p12T24 1 - 34 T + 274 T^{2} + 1008 T^{3} + 24830 T^{4} - 899081 T^{5} + 3408808 T^{6} + 17123755 T^{7} + 654386474 T^{8} - 7963303748 T^{9} - 4308732765 T^{10} - 202867350615 T^{11} + 6976154776464 T^{12} - 202867350615 p T^{13} - 4308732765 p^{2} T^{14} - 7963303748 p^{3} T^{15} + 654386474 p^{4} T^{16} + 17123755 p^{5} T^{17} + 3408808 p^{6} T^{18} - 899081 p^{7} T^{19} + 24830 p^{8} T^{20} + 1008 p^{9} T^{21} + 274 p^{10} T^{22} - 34 p^{11} T^{23} + p^{12} T^{24}
83 1−44T+709T2−6594T3+101865T4−1690345T5+14037545T6−106393506T7+1871755556T8−19587962148T9+104769987163T10−1328892312861T11+18520826492279T12−1328892312861pT13+104769987163p2T14−19587962148p3T15+1871755556p4T16−106393506p5T17+14037545p6T18−1690345p7T19+101865p8T20−6594p9T21+709p10T22−44p11T23+p12T24 1 - 44 T + 709 T^{2} - 6594 T^{3} + 101865 T^{4} - 1690345 T^{5} + 14037545 T^{6} - 106393506 T^{7} + 1871755556 T^{8} - 19587962148 T^{9} + 104769987163 T^{10} - 1328892312861 T^{11} + 18520826492279 T^{12} - 1328892312861 p T^{13} + 104769987163 p^{2} T^{14} - 19587962148 p^{3} T^{15} + 1871755556 p^{4} T^{16} - 106393506 p^{5} T^{17} + 14037545 p^{6} T^{18} - 1690345 p^{7} T^{19} + 101865 p^{8} T^{20} - 6594 p^{9} T^{21} + 709 p^{10} T^{22} - 44 p^{11} T^{23} + p^{12} T^{24}
89 (1−11T+243T2−2974T3+41947T4−395342T5+4515440T6−395342pT7+41947p2T8−2974p3T9+243p4T10−11p5T11+p6T12)2 ( 1 - 11 T + 243 T^{2} - 2974 T^{3} + 41947 T^{4} - 395342 T^{5} + 4515440 T^{6} - 395342 p T^{7} + 41947 p^{2} T^{8} - 2974 p^{3} T^{9} + 243 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2}
97 1−12T−350T2+3834T3+79315T4−724890T5−12688285T6+91701159T7+1583812456T8−7590111246T9−170176594430T10+298340892537T11+16744508198807T12+298340892537pT13−170176594430p2T14−7590111246p3T15+1583812456p4T16+91701159p5T17−12688285p6T18−724890p7T19+79315p8T20+3834p9T21−350p10T22−12p11T23+p12T24 1 - 12 T - 350 T^{2} + 3834 T^{3} + 79315 T^{4} - 724890 T^{5} - 12688285 T^{6} + 91701159 T^{7} + 1583812456 T^{8} - 7590111246 T^{9} - 170176594430 T^{10} + 298340892537 T^{11} + 16744508198807 T^{12} + 298340892537 p T^{13} - 170176594430 p^{2} T^{14} - 7590111246 p^{3} T^{15} + 1583812456 p^{4} T^{16} + 91701159 p^{5} T^{17} - 12688285 p^{6} T^{18} - 724890 p^{7} T^{19} + 79315 p^{8} T^{20} + 3834 p^{9} T^{21} - 350 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24}
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   L(s)=∏p ∏j=124(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.33858123952618020793268587630, −3.19867428819935466716384318663, −2.94032196507401585094190844429, −2.93945740696852761018596005840, −2.81662922118837029766478487300, −2.77558778922723905447479531265, −2.67214527728549115875220848834, −2.44038252360750651613073582100, −2.35532120679122174166424388172, −2.34099311866090706441336826909, −2.24315646565439912957307637623, −2.13928582058603408450679116973, −1.91854730772275985281308253331, −1.80885900056585596825402851655, −1.72396290357897732497574708652, −1.61178769264283595394519327053, −1.53408320983449606364415441197, −1.14452602568307500266244192473, −1.06565837781319769034503606390, −0.926647455697108414610529628188, −0.78438853741718652349538564293, −0.77705436140159678214966653067, −0.75813660728091640594898891803, −0.51359444185754010523978858435, −0.12938571558753131887441652676, 0.12938571558753131887441652676, 0.51359444185754010523978858435, 0.75813660728091640594898891803, 0.77705436140159678214966653067, 0.78438853741718652349538564293, 0.926647455697108414610529628188, 1.06565837781319769034503606390, 1.14452602568307500266244192473, 1.53408320983449606364415441197, 1.61178769264283595394519327053, 1.72396290357897732497574708652, 1.80885900056585596825402851655, 1.91854730772275985281308253331, 2.13928582058603408450679116973, 2.24315646565439912957307637623, 2.34099311866090706441336826909, 2.35532120679122174166424388172, 2.44038252360750651613073582100, 2.67214527728549115875220848834, 2.77558778922723905447479531265, 2.81662922118837029766478487300, 2.93945740696852761018596005840, 2.94032196507401585094190844429, 3.19867428819935466716384318663, 3.33858123952618020793268587630

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.