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Properties

Label 24-368e12-1.1-c1e12-0-1
Degree 2424
Conductor 6.168×10306.168\times 10^{30}
Sign 11
Analytic cond. 414475.414475.
Root an. cond. 1.714201.71420
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  + 2·2-s + 2·3-s + 2·4-s − 4·5-s + 4·6-s + 2·9-s − 8·10-s − 4·11-s + 4·12-s + 18·13-s − 8·15-s − 8·17-s + 4·18-s − 8·19-s − 8·20-s − 8·22-s + 8·25-s + 36·26-s + 8·27-s + 2·29-s − 16·30-s + 20·31-s − 8·33-s − 16·34-s + 4·36-s − 4·37-s − 16·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s + 2/3·9-s − 2.52·10-s − 1.20·11-s + 1.15·12-s + 4.99·13-s − 2.06·15-s − 1.94·17-s + 0.942·18-s − 1.83·19-s − 1.78·20-s − 1.70·22-s + 8/5·25-s + 7.06·26-s + 1.53·27-s + 0.371·29-s − 2.92·30-s + 3.59·31-s − 1.39·33-s − 2.74·34-s + 2/3·36-s − 0.657·37-s − 2.59·38-s + ⋯

Functional equation

Λ(s)=((248⋅2312)s/2 ΓC(s)12 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((248⋅2312)s/2 ΓC(s+1/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 248⋅23122^{48} \cdot 23^{12}
Sign: 11
Analytic conductor: 414475.414475.
Root analytic conductor: 1.714201.71420
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 248⋅2312, ( :[1/2]12), 1)(24,\ 2^{48} \cdot 23^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 22.4240985122.42409851
L(12)L(\frac12) ≈\approx 22.4240985122.42409851
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−pT+pT2−p2T4+p3T5−3p2T6+p4T7−p4T8+p5T10−p6T11+p6T12 1 - p T + p T^{2} - p^{2} T^{4} + p^{3} T^{5} - 3 p^{2} T^{6} + p^{4} T^{7} - p^{4} T^{8} + p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12}
23 (1+T2)6 ( 1 + T^{2} )^{6}
good3 1−2T+2T2−8T3+5T4+20T5−2p2T6+70T7−190T8+2p3T9+2T10−176T11+1273T12−176pT13+2p2T14+2p6T15−190p4T16+70p5T17−2p8T18+20p7T19+5p8T20−8p9T21+2p10T22−2p11T23+p12T24 1 - 2 T + 2 T^{2} - 8 T^{3} + 5 T^{4} + 20 T^{5} - 2 p^{2} T^{6} + 70 T^{7} - 190 T^{8} + 2 p^{3} T^{9} + 2 T^{10} - 176 T^{11} + 1273 T^{12} - 176 p T^{13} + 2 p^{2} T^{14} + 2 p^{6} T^{15} - 190 p^{4} T^{16} + 70 p^{5} T^{17} - 2 p^{8} T^{18} + 20 p^{7} T^{19} + 5 p^{8} T^{20} - 8 p^{9} T^{21} + 2 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24}
5 1+4T+8T2−78T4−212T5−224T6+352T7+2731T8+4604T9+1568T10−17028T11−69976T12−17028pT13+1568p2T14+4604p3T15+2731p4T16+352p5T17−224p6T18−212p7T19−78p8T20+8p10T22+4p11T23+p12T24 1 + 4 T + 8 T^{2} - 78 T^{4} - 212 T^{5} - 224 T^{6} + 352 T^{7} + 2731 T^{8} + 4604 T^{9} + 1568 T^{10} - 17028 T^{11} - 69976 T^{12} - 17028 p T^{13} + 1568 p^{2} T^{14} + 4604 p^{3} T^{15} + 2731 p^{4} T^{16} + 352 p^{5} T^{17} - 224 p^{6} T^{18} - 212 p^{7} T^{19} - 78 p^{8} T^{20} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24}
7 1−68T2+2206T4−45092T6+13191p2T8−977464pT10+54838648T12−977464p3T14+13191p6T16−45092p6T18+2206p8T20−68p10T22+p12T24 1 - 68 T^{2} + 2206 T^{4} - 45092 T^{6} + 13191 p^{2} T^{8} - 977464 p T^{10} + 54838648 T^{12} - 977464 p^{3} T^{14} + 13191 p^{6} T^{16} - 45092 p^{6} T^{18} + 2206 p^{8} T^{20} - 68 p^{10} T^{22} + p^{12} T^{24}
11 1+4T+8T2+52T3−134T4−1140T5−2136T6−13484T7−22433T8+75376T9+143376T10+912056T11+5810608T12+912056pT13+143376p2T14+75376p3T15−22433p4T16−13484p5T17−2136p6T18−1140p7T19−134p8T20+52p9T21+8p10T22+4p11T23+p12T24 1 + 4 T + 8 T^{2} + 52 T^{3} - 134 T^{4} - 1140 T^{5} - 2136 T^{6} - 13484 T^{7} - 22433 T^{8} + 75376 T^{9} + 143376 T^{10} + 912056 T^{11} + 5810608 T^{12} + 912056 p T^{13} + 143376 p^{2} T^{14} + 75376 p^{3} T^{15} - 22433 p^{4} T^{16} - 13484 p^{5} T^{17} - 2136 p^{6} T^{18} - 1140 p^{7} T^{19} - 134 p^{8} T^{20} + 52 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24}
13 1−18T+162T2−84pT3+6673T4−37448T5+189270T6−889826T7+3998178T8−17024206T9+68598506T10−265184208T11+981270081T12−265184208pT13+68598506p2T14−17024206p3T15+3998178p4T16−889826p5T17+189270p6T18−37448p7T19+6673p8T20−84p10T21+162p10T22−18p11T23+p12T24 1 - 18 T + 162 T^{2} - 84 p T^{3} + 6673 T^{4} - 37448 T^{5} + 189270 T^{6} - 889826 T^{7} + 3998178 T^{8} - 17024206 T^{9} + 68598506 T^{10} - 265184208 T^{11} + 981270081 T^{12} - 265184208 p T^{13} + 68598506 p^{2} T^{14} - 17024206 p^{3} T^{15} + 3998178 p^{4} T^{16} - 889826 p^{5} T^{17} + 189270 p^{6} T^{18} - 37448 p^{7} T^{19} + 6673 p^{8} T^{20} - 84 p^{10} T^{21} + 162 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24}
17 (1+4T+64T2+246T3+2065T4+430pT5+43246T6+430p2T7+2065p2T8+246p3T9+64p4T10+4p5T11+p6T12)2 ( 1 + 4 T + 64 T^{2} + 246 T^{3} + 2065 T^{4} + 430 p T^{5} + 43246 T^{6} + 430 p^{2} T^{7} + 2065 p^{2} T^{8} + 246 p^{3} T^{9} + 64 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2}
19 1+8T+32T2+8pT3+1418T4+8680T5+35616T6+179896T7+1102335T8+5132496T9+20297152T10+101957616T11+511158412T12+101957616pT13+20297152p2T14+5132496p3T15+1102335p4T16+179896p5T17+35616p6T18+8680p7T19+1418p8T20+8p10T21+32p10T22+8p11T23+p12T24 1 + 8 T + 32 T^{2} + 8 p T^{3} + 1418 T^{4} + 8680 T^{5} + 35616 T^{6} + 179896 T^{7} + 1102335 T^{8} + 5132496 T^{9} + 20297152 T^{10} + 101957616 T^{11} + 511158412 T^{12} + 101957616 p T^{13} + 20297152 p^{2} T^{14} + 5132496 p^{3} T^{15} + 1102335 p^{4} T^{16} + 179896 p^{5} T^{17} + 35616 p^{6} T^{18} + 8680 p^{7} T^{19} + 1418 p^{8} T^{20} + 8 p^{10} T^{21} + 32 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24}
29 1−2T+2T2−424T3+2329T4+1780T5+81670T6−841602T7+1013250T8−114454pT9+172825850T10−584430420T11−337880919T12−584430420pT13+172825850p2T14−114454p4T15+1013250p4T16−841602p5T17+81670p6T18+1780p7T19+2329p8T20−424p9T21+2p10T22−2p11T23+p12T24 1 - 2 T + 2 T^{2} - 424 T^{3} + 2329 T^{4} + 1780 T^{5} + 81670 T^{6} - 841602 T^{7} + 1013250 T^{8} - 114454 p T^{9} + 172825850 T^{10} - 584430420 T^{11} - 337880919 T^{12} - 584430420 p T^{13} + 172825850 p^{2} T^{14} - 114454 p^{4} T^{15} + 1013250 p^{4} T^{16} - 841602 p^{5} T^{17} + 81670 p^{6} T^{18} + 1780 p^{7} T^{19} + 2329 p^{8} T^{20} - 424 p^{9} T^{21} + 2 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24}
31 (1−10T+143T2−1220T3+9818T4−67262T5+389725T6−67262pT7+9818p2T8−1220p3T9+143p4T10−10p5T11+p6T12)2 ( 1 - 10 T + 143 T^{2} - 1220 T^{3} + 9818 T^{4} - 67262 T^{5} + 389725 T^{6} - 67262 p T^{7} + 9818 p^{2} T^{8} - 1220 p^{3} T^{9} + 143 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2}
37 1+4T+8T2+232T3+14pT4−9756T5−16256T6−371240T7−61273pT8+5079796T9+8338688T10+230905436T11+6813255824T12+230905436pT13+8338688p2T14+5079796p3T15−61273p5T16−371240p5T17−16256p6T18−9756p7T19+14p9T20+232p9T21+8p10T22+4p11T23+p12T24 1 + 4 T + 8 T^{2} + 232 T^{3} + 14 p T^{4} - 9756 T^{5} - 16256 T^{6} - 371240 T^{7} - 61273 p T^{8} + 5079796 T^{9} + 8338688 T^{10} + 230905436 T^{11} + 6813255824 T^{12} + 230905436 p T^{13} + 8338688 p^{2} T^{14} + 5079796 p^{3} T^{15} - 61273 p^{5} T^{16} - 371240 p^{5} T^{17} - 16256 p^{6} T^{18} - 9756 p^{7} T^{19} + 14 p^{9} T^{20} + 232 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24}
41 1−254T2+32757T4−2923858T6+201141458T8−11090716582T10+500464443061T12−11090716582p2T14+201141458p4T16−2923858p6T18+32757p8T20−254p10T22+p12T24 1 - 254 T^{2} + 32757 T^{4} - 2923858 T^{6} + 201141458 T^{8} - 11090716582 T^{10} + 500464443061 T^{12} - 11090716582 p^{2} T^{14} + 201141458 p^{4} T^{16} - 2923858 p^{6} T^{18} + 32757 p^{8} T^{20} - 254 p^{10} T^{22} + p^{12} T^{24}
43 1−20T+200T2−924T3−1770T4+46524T5−149592T6−1971628T7+27832799T8−162350984T9+319707600T10+2960692776T11−33765637196T12+2960692776pT13+319707600p2T14−162350984p3T15+27832799p4T16−1971628p5T17−149592p6T18+46524p7T19−1770p8T20−924p9T21+200p10T22−20p11T23+p12T24 1 - 20 T + 200 T^{2} - 924 T^{3} - 1770 T^{4} + 46524 T^{5} - 149592 T^{6} - 1971628 T^{7} + 27832799 T^{8} - 162350984 T^{9} + 319707600 T^{10} + 2960692776 T^{11} - 33765637196 T^{12} + 2960692776 p T^{13} + 319707600 p^{2} T^{14} - 162350984 p^{3} T^{15} + 27832799 p^{4} T^{16} - 1971628 p^{5} T^{17} - 149592 p^{6} T^{18} + 46524 p^{7} T^{19} - 1770 p^{8} T^{20} - 924 p^{9} T^{21} + 200 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24}
47 (1+8T+147T2+830T3+11506T4+50068T5+588553T6+50068pT7+11506p2T8+830p3T9+147p4T10+8p5T11+p6T12)2 ( 1 + 8 T + 147 T^{2} + 830 T^{3} + 11506 T^{4} + 50068 T^{5} + 588553 T^{6} + 50068 p T^{7} + 11506 p^{2} T^{8} + 830 p^{3} T^{9} + 147 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2}
53 1+16T+128T2+1096T3+586T4−82200T5−789600T6−7806032T7−45654497T8+14272960T9+1357857504T10+20288254256T11+219339685132T12+20288254256pT13+1357857504p2T14+14272960p3T15−45654497p4T16−7806032p5T17−789600p6T18−82200p7T19+586p8T20+1096p9T21+128p10T22+16p11T23+p12T24 1 + 16 T + 128 T^{2} + 1096 T^{3} + 586 T^{4} - 82200 T^{5} - 789600 T^{6} - 7806032 T^{7} - 45654497 T^{8} + 14272960 T^{9} + 1357857504 T^{10} + 20288254256 T^{11} + 219339685132 T^{12} + 20288254256 p T^{13} + 1357857504 p^{2} T^{14} + 14272960 p^{3} T^{15} - 45654497 p^{4} T^{16} - 7806032 p^{5} T^{17} - 789600 p^{6} T^{18} - 82200 p^{7} T^{19} + 586 p^{8} T^{20} + 1096 p^{9} T^{21} + 128 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24}
59 1−8T+32T2+312T3−8214T4+37896T5+8352T6−1753848T7+32803743T8−128318160T9+216281664T10+6242043056T11−111681131956T12+6242043056pT13+216281664p2T14−128318160p3T15+32803743p4T16−1753848p5T17+8352p6T18+37896p7T19−8214p8T20+312p9T21+32p10T22−8p11T23+p12T24 1 - 8 T + 32 T^{2} + 312 T^{3} - 8214 T^{4} + 37896 T^{5} + 8352 T^{6} - 1753848 T^{7} + 32803743 T^{8} - 128318160 T^{9} + 216281664 T^{10} + 6242043056 T^{11} - 111681131956 T^{12} + 6242043056 p T^{13} + 216281664 p^{2} T^{14} - 128318160 p^{3} T^{15} + 32803743 p^{4} T^{16} - 1753848 p^{5} T^{17} + 8352 p^{6} T^{18} + 37896 p^{7} T^{19} - 8214 p^{8} T^{20} + 312 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24}
61 1−12T+72T2−436T3+10182T4−96692T5+522248T6−962348T7−2556417T8−61735480T9+701613808T10+10527482232T11−198377340076T12+10527482232pT13+701613808p2T14−61735480p3T15−2556417p4T16−962348p5T17+522248p6T18−96692p7T19+10182p8T20−436p9T21+72p10T22−12p11T23+p12T24 1 - 12 T + 72 T^{2} - 436 T^{3} + 10182 T^{4} - 96692 T^{5} + 522248 T^{6} - 962348 T^{7} - 2556417 T^{8} - 61735480 T^{9} + 701613808 T^{10} + 10527482232 T^{11} - 198377340076 T^{12} + 10527482232 p T^{13} + 701613808 p^{2} T^{14} - 61735480 p^{3} T^{15} - 2556417 p^{4} T^{16} - 962348 p^{5} T^{17} + 522248 p^{6} T^{18} - 96692 p^{7} T^{19} + 10182 p^{8} T^{20} - 436 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24}
67 1+4T+8T2+76T3−1562T4+12052T5+63592T6+2247636T7−19830089T8−229054608T9−414801328T10−5101901080T11+60684024232T12−5101901080pT13−414801328p2T14−229054608p3T15−19830089p4T16+2247636p5T17+63592p6T18+12052p7T19−1562p8T20+76p9T21+8p10T22+4p11T23+p12T24 1 + 4 T + 8 T^{2} + 76 T^{3} - 1562 T^{4} + 12052 T^{5} + 63592 T^{6} + 2247636 T^{7} - 19830089 T^{8} - 229054608 T^{9} - 414801328 T^{10} - 5101901080 T^{11} + 60684024232 T^{12} - 5101901080 p T^{13} - 414801328 p^{2} T^{14} - 229054608 p^{3} T^{15} - 19830089 p^{4} T^{16} + 2247636 p^{5} T^{17} + 63592 p^{6} T^{18} + 12052 p^{7} T^{19} - 1562 p^{8} T^{20} + 76 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24}
71 1−586T2+166561T4−30467270T6+4018404002T8−405369799706T10+32251408969501T12−405369799706p2T14+4018404002p4T16−30467270p6T18+166561p8T20−586p10T22+p12T24 1 - 586 T^{2} + 166561 T^{4} - 30467270 T^{6} + 4018404002 T^{8} - 405369799706 T^{10} + 32251408969501 T^{12} - 405369799706 p^{2} T^{14} + 4018404002 p^{4} T^{16} - 30467270 p^{6} T^{18} + 166561 p^{8} T^{20} - 586 p^{10} T^{22} + p^{12} T^{24}
73 1−422T2+93189T4−13924530T6+1577319194T8−145394149854T10+11405608578077T12−145394149854p2T14+1577319194p4T16−13924530p6T18+93189p8T20−422p10T22+p12T24 1 - 422 T^{2} + 93189 T^{4} - 13924530 T^{6} + 1577319194 T^{8} - 145394149854 T^{10} + 11405608578077 T^{12} - 145394149854 p^{2} T^{14} + 1577319194 p^{4} T^{16} - 13924530 p^{6} T^{18} + 93189 p^{8} T^{20} - 422 p^{10} T^{22} + p^{12} T^{24}
79 (1+2T+368T2+1126T3+60155T4+204124T5+5897072T6+204124pT7+60155p2T8+1126p3T9+368p4T10+2p5T11+p6T12)2 ( 1 + 2 T + 368 T^{2} + 1126 T^{3} + 60155 T^{4} + 204124 T^{5} + 5897072 T^{6} + 204124 p T^{7} + 60155 p^{2} T^{8} + 1126 p^{3} T^{9} + 368 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2}
83 1−28T+392T2−5876T3+84726T4−894300T5+9091496T6−99906012T7+883151735T8−7070500512T9+68426415632T10−574097189672T11+4396159810696T12−574097189672pT13+68426415632p2T14−7070500512p3T15+883151735p4T16−99906012p5T17+9091496p6T18−894300p7T19+84726p8T20−5876p9T21+392p10T22−28p11T23+p12T24 1 - 28 T + 392 T^{2} - 5876 T^{3} + 84726 T^{4} - 894300 T^{5} + 9091496 T^{6} - 99906012 T^{7} + 883151735 T^{8} - 7070500512 T^{9} + 68426415632 T^{10} - 574097189672 T^{11} + 4396159810696 T^{12} - 574097189672 p T^{13} + 68426415632 p^{2} T^{14} - 7070500512 p^{3} T^{15} + 883151735 p^{4} T^{16} - 99906012 p^{5} T^{17} + 9091496 p^{6} T^{18} - 894300 p^{7} T^{19} + 84726 p^{8} T^{20} - 5876 p^{9} T^{21} + 392 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24}
89 1−636T2+200302T4−41757740T6+6485840655T8−793818201688T10+78404791964292T12−793818201688p2T14+6485840655p4T16−41757740p6T18+200302p8T20−636p10T22+p12T24 1 - 636 T^{2} + 200302 T^{4} - 41757740 T^{6} + 6485840655 T^{8} - 793818201688 T^{10} + 78404791964292 T^{12} - 793818201688 p^{2} T^{14} + 6485840655 p^{4} T^{16} - 41757740 p^{6} T^{18} + 200302 p^{8} T^{20} - 636 p^{10} T^{22} + p^{12} T^{24}
97 (1−18T+446T2−6848T3+95053T4−1194926T5+11749682T6−1194926pT7+95053p2T8−6848p3T9+446p4T10−18p5T11+p6T12)2 ( 1 - 18 T + 446 T^{2} - 6848 T^{3} + 95053 T^{4} - 1194926 T^{5} + 11749682 T^{6} - 1194926 p T^{7} + 95053 p^{2} T^{8} - 6848 p^{3} T^{9} + 446 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2}
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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

−3.87557437151970181617503925616, −3.83606431292845019629782321294, −3.81736409660597829717095700935, −3.71695112095465242551714340924, −3.45007524292191615862839947186, −3.41355724765996220278334269970, −3.17806668114539975544731516127, −3.09173358998269879976215077889, −3.02976387776582137422094243857, −2.98487059449252050967082662796, −2.74559376709046905725743728081, −2.63362742949133578597468971440, −2.40091883385074974794375894198, −2.39511395399818928211086739556, −2.33977441100508406934107123901, −2.14647611322891899249383761149, −2.09662749487150721203025885434, −1.89793299546158056440045377543, −1.89314453486748937723889002260, −1.24237843368814889930926872295, −1.12437948643308533121759092715, −0.983335368265266889618073751129, −0.981301979300453239216381280897, −0.76512604202615608171771781712, −0.51860020902536311059119138090, 0.51860020902536311059119138090, 0.76512604202615608171771781712, 0.981301979300453239216381280897, 0.983335368265266889618073751129, 1.12437948643308533121759092715, 1.24237843368814889930926872295, 1.89314453486748937723889002260, 1.89793299546158056440045377543, 2.09662749487150721203025885434, 2.14647611322891899249383761149, 2.33977441100508406934107123901, 2.39511395399818928211086739556, 2.40091883385074974794375894198, 2.63362742949133578597468971440, 2.74559376709046905725743728081, 2.98487059449252050967082662796, 3.02976387776582137422094243857, 3.09173358998269879976215077889, 3.17806668114539975544731516127, 3.41355724765996220278334269970, 3.45007524292191615862839947186, 3.71695112095465242551714340924, 3.81736409660597829717095700935, 3.83606431292845019629782321294, 3.87557437151970181617503925616

Graph of the ZZ-function along the critical line

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