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Properties

Label 24-637e12-1.1-c1e12-0-8
Degree 2424
Conductor 4.463×10334.463\times 10^{33}
Sign 11
Analytic cond. 2.99915×1082.99915\times 10^{8}
Root an. cond. 2.255322.25532
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  − 4·2-s − 3-s − 5-s + 4·6-s + 22·8-s + 11·9-s + 4·10-s + 4·11-s + 2·13-s + 15-s − 32·16-s + 10·17-s − 44·18-s + 19-s − 16·22-s + 2·23-s − 22·24-s + 19·25-s − 8·26-s − 10·27-s + 3·29-s − 4·30-s − 16·31-s − 20·32-s − 4·33-s − 40·34-s + 26·37-s + ⋯
L(s)  = 1  − 2.82·2-s − 0.577·3-s − 0.447·5-s + 1.63·6-s + 7.77·8-s + 11/3·9-s + 1.26·10-s + 1.20·11-s + 0.554·13-s + 0.258·15-s − 8·16-s + 2.42·17-s − 10.3·18-s + 0.229·19-s − 3.41·22-s + 0.417·23-s − 4.49·24-s + 19/5·25-s − 1.56·26-s − 1.92·27-s + 0.557·29-s − 0.730·30-s − 2.87·31-s − 3.53·32-s − 0.696·33-s − 6.85·34-s + 4.27·37-s + ⋯

Functional equation

Λ(s)=((724⋅1312)s/2 ΓC(s)12 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((724⋅1312)s/2 ΓC(s+1/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{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: 724⋅13127^{24} \cdot 13^{12}
Sign: 11
Analytic conductor: 2.99915×1082.99915\times 10^{8}
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 724⋅1312, ( :[1/2]12), 1)(24,\ 7^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.44175676600.4417567660
L(12)L(\frac12) ≈\approx 0.44175676600.4417567660
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)
bad7 1 1
13 1−2T−16T2−3T3+607T4−433T5−5615T6−433pT7+607p2T8−3p3T9−16p4T10−2p5T11+p6T12 1 - 2 T - 16 T^{2} - 3 T^{3} + 607 T^{4} - 433 T^{5} - 5615 T^{6} - 433 p T^{7} + 607 p^{2} T^{8} - 3 p^{3} T^{9} - 16 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
good2 (1+pT+3pT2+9T3+5p2T4+7p2T5+51T6+7p3T7+5p4T8+9p3T9+3p5T10+p6T11+p6T12)2 ( 1 + p T + 3 p T^{2} + 9 T^{3} + 5 p^{2} T^{4} + 7 p^{2} T^{5} + 51 T^{6} + 7 p^{3} T^{7} + 5 p^{4} T^{8} + 9 p^{3} T^{9} + 3 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2}
3 1+T−10T2−11T3+53T4+62T5−167T6−221T7+98pT8+535T9+79T10−604T11−1559T12−604pT13+79p2T14+535p3T15+98p5T16−221p5T17−167p6T18+62p7T19+53p8T20−11p9T21−10p10T22+p11T23+p12T24 1 + T - 10 T^{2} - 11 T^{3} + 53 T^{4} + 62 T^{5} - 167 T^{6} - 221 T^{7} + 98 p T^{8} + 535 T^{9} + 79 T^{10} - 604 T^{11} - 1559 T^{12} - 604 p T^{13} + 79 p^{2} T^{14} + 535 p^{3} T^{15} + 98 p^{5} T^{16} - 221 p^{5} T^{17} - 167 p^{6} T^{18} + 62 p^{7} T^{19} + 53 p^{8} T^{20} - 11 p^{9} T^{21} - 10 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24}
5 1+T−18T2+pT3+193T4−192T5−1181T6+2139T7+908pT8−451p2T9−6679T10+26266T11+249T12+26266pT13−6679p2T14−451p5T15+908p5T16+2139p5T17−1181p6T18−192p7T19+193p8T20+p10T21−18p10T22+p11T23+p12T24 1 + T - 18 T^{2} + p T^{3} + 193 T^{4} - 192 T^{5} - 1181 T^{6} + 2139 T^{7} + 908 p T^{8} - 451 p^{2} T^{9} - 6679 T^{10} + 26266 T^{11} + 249 T^{12} + 26266 p T^{13} - 6679 p^{2} T^{14} - 451 p^{5} T^{15} + 908 p^{5} T^{16} + 2139 p^{5} T^{17} - 1181 p^{6} T^{18} - 192 p^{7} T^{19} + 193 p^{8} T^{20} + p^{10} T^{21} - 18 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24}
11 1−4T−29T2+108T3+477T4−113pT5−6686T6+7665T7+89323T8+423T9−1282040T10−249219T11+16505087T12−249219pT13−1282040p2T14+423p3T15+89323p4T16+7665p5T17−6686p6T18−113p8T19+477p8T20+108p9T21−29p10T22−4p11T23+p12T24 1 - 4 T - 29 T^{2} + 108 T^{3} + 477 T^{4} - 113 p T^{5} - 6686 T^{6} + 7665 T^{7} + 89323 T^{8} + 423 T^{9} - 1282040 T^{10} - 249219 T^{11} + 16505087 T^{12} - 249219 p T^{13} - 1282040 p^{2} T^{14} + 423 p^{3} T^{15} + 89323 p^{4} T^{16} + 7665 p^{5} T^{17} - 6686 p^{6} T^{18} - 113 p^{8} T^{19} + 477 p^{8} T^{20} + 108 p^{9} T^{21} - 29 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24}
17 (1−5T+90T2−411T3+3539T4−13744T5+78123T6−13744pT7+3539p2T8−411p3T9+90p4T10−5p5T11+p6T12)2 ( 1 - 5 T + 90 T^{2} - 411 T^{3} + 3539 T^{4} - 13744 T^{5} + 78123 T^{6} - 13744 p T^{7} + 3539 p^{2} T^{8} - 411 p^{3} T^{9} + 90 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2}
19 1−T−49T2−82T3+1336T4+4335T5−10907T6−99626T7−263580T8+1110690T9+9684539T10−2414194T11−215227743T12−2414194pT13+9684539p2T14+1110690p3T15−263580p4T16−99626p5T17−10907p6T18+4335p7T19+1336p8T20−82p9T21−49p10T22−p11T23+p12T24 1 - T - 49 T^{2} - 82 T^{3} + 1336 T^{4} + 4335 T^{5} - 10907 T^{6} - 99626 T^{7} - 263580 T^{8} + 1110690 T^{9} + 9684539 T^{10} - 2414194 T^{11} - 215227743 T^{12} - 2414194 p T^{13} + 9684539 p^{2} T^{14} + 1110690 p^{3} T^{15} - 263580 p^{4} T^{16} - 99626 p^{5} T^{17} - 10907 p^{6} T^{18} + 4335 p^{7} T^{19} + 1336 p^{8} T^{20} - 82 p^{9} T^{21} - 49 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24}
23 (1−T+32T2−52T3+1214T4−1383T5+21935T6−1383pT7+1214p2T8−52p3T9+32p4T10−p5T11+p6T12)2 ( 1 - T + 32 T^{2} - 52 T^{3} + 1214 T^{4} - 1383 T^{5} + 21935 T^{6} - 1383 p T^{7} + 1214 p^{2} T^{8} - 52 p^{3} T^{9} + 32 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2}
29 1−3T−3pT2+94T3+158pT4+904T5−123467T6−308042T7+1215550T8+10832851T9+73693658T10−174959016T11−3324017493T12−174959016pT13+73693658p2T14+10832851p3T15+1215550p4T16−308042p5T17−123467p6T18+904p7T19+158p9T20+94p9T21−3p11T22−3p11T23+p12T24 1 - 3 T - 3 p T^{2} + 94 T^{3} + 158 p T^{4} + 904 T^{5} - 123467 T^{6} - 308042 T^{7} + 1215550 T^{8} + 10832851 T^{9} + 73693658 T^{10} - 174959016 T^{11} - 3324017493 T^{12} - 174959016 p T^{13} + 73693658 p^{2} T^{14} + 10832851 p^{3} T^{15} + 1215550 p^{4} T^{16} - 308042 p^{5} T^{17} - 123467 p^{6} T^{18} + 904 p^{7} T^{19} + 158 p^{9} T^{20} + 94 p^{9} T^{21} - 3 p^{11} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24}
31 1+16T+20T2−594T3+2163T4+43649T5−56125T6−1282696T7+2984747T8+22743273T9−180497697T10−655302586T11+2182678017T12−655302586pT13−180497697p2T14+22743273p3T15+2984747p4T16−1282696p5T17−56125p6T18+43649p7T19+2163p8T20−594p9T21+20p10T22+16p11T23+p12T24 1 + 16 T + 20 T^{2} - 594 T^{3} + 2163 T^{4} + 43649 T^{5} - 56125 T^{6} - 1282696 T^{7} + 2984747 T^{8} + 22743273 T^{9} - 180497697 T^{10} - 655302586 T^{11} + 2182678017 T^{12} - 655302586 p T^{13} - 180497697 p^{2} T^{14} + 22743273 p^{3} T^{15} + 2984747 p^{4} T^{16} - 1282696 p^{5} T^{17} - 56125 p^{6} T^{18} + 43649 p^{7} T^{19} + 2163 p^{8} T^{20} - 594 p^{9} T^{21} + 20 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24}
37 (1−13T+184T2−1054T3+7158T4−10573T5+113729T6−10573pT7+7158p2T8−1054p3T9+184p4T10−13p5T11+p6T12)2 ( 1 - 13 T + 184 T^{2} - 1054 T^{3} + 7158 T^{4} - 10573 T^{5} + 113729 T^{6} - 10573 p T^{7} + 7158 p^{2} T^{8} - 1054 p^{3} T^{9} + 184 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )^{2}
41 1−8T−161T2+924T3+18241T4−64367T5−1502654T6+3175261T7+96068491T8−105301221T9−5078164754T10+1647875431T11+226350132753T12+1647875431pT13−5078164754p2T14−105301221p3T15+96068491p4T16+3175261p5T17−1502654p6T18−64367p7T19+18241p8T20+924p9T21−161p10T22−8p11T23+p12T24 1 - 8 T - 161 T^{2} + 924 T^{3} + 18241 T^{4} - 64367 T^{5} - 1502654 T^{6} + 3175261 T^{7} + 96068491 T^{8} - 105301221 T^{9} - 5078164754 T^{10} + 1647875431 T^{11} + 226350132753 T^{12} + 1647875431 p T^{13} - 5078164754 p^{2} T^{14} - 105301221 p^{3} T^{15} + 96068491 p^{4} T^{16} + 3175261 p^{5} T^{17} - 1502654 p^{6} T^{18} - 64367 p^{7} T^{19} + 18241 p^{8} T^{20} + 924 p^{9} T^{21} - 161 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24}
43 1+11T−138T2−1349T3+16370T4+106653T5−1472431T6−5757651T7+106708219T8+224797058T9−6088028976T10−3777766292T11+288640495545T12−3777766292pT13−6088028976p2T14+224797058p3T15+106708219p4T16−5757651p5T17−1472431p6T18+106653p7T19+16370p8T20−1349p9T21−138p10T22+11p11T23+p12T24 1 + 11 T - 138 T^{2} - 1349 T^{3} + 16370 T^{4} + 106653 T^{5} - 1472431 T^{6} - 5757651 T^{7} + 106708219 T^{8} + 224797058 T^{9} - 6088028976 T^{10} - 3777766292 T^{11} + 288640495545 T^{12} - 3777766292 p T^{13} - 6088028976 p^{2} T^{14} + 224797058 p^{3} T^{15} + 106708219 p^{4} T^{16} - 5757651 p^{5} T^{17} - 1472431 p^{6} T^{18} + 106653 p^{7} T^{19} + 16370 p^{8} T^{20} - 1349 p^{9} T^{21} - 138 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24}
47 1−T−104T2+189T3+5335T4−164pT5−69863T6−514255T7−7627520T8+55687467T9+662939941T10−1686387922T11−35399065407T12−1686387922pT13+662939941p2T14+55687467p3T15−7627520p4T16−514255p5T17−69863p6T18−164p8T19+5335p8T20+189p9T21−104p10T22−p11T23+p12T24 1 - T - 104 T^{2} + 189 T^{3} + 5335 T^{4} - 164 p T^{5} - 69863 T^{6} - 514255 T^{7} - 7627520 T^{8} + 55687467 T^{9} + 662939941 T^{10} - 1686387922 T^{11} - 35399065407 T^{12} - 1686387922 p T^{13} + 662939941 p^{2} T^{14} + 55687467 p^{3} T^{15} - 7627520 p^{4} T^{16} - 514255 p^{5} T^{17} - 69863 p^{6} T^{18} - 164 p^{8} T^{19} + 5335 p^{8} T^{20} + 189 p^{9} T^{21} - 104 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24}
53 1+2T−214T2−252T3+24796T4+13772T5−1921862T6+82142T7+113089342T8−43114584T9−5653831794T10+1443208718T11+285781391787T12+1443208718pT13−5653831794p2T14−43114584p3T15+113089342p4T16+82142p5T17−1921862p6T18+13772p7T19+24796p8T20−252p9T21−214p10T22+2p11T23+p12T24 1 + 2 T - 214 T^{2} - 252 T^{3} + 24796 T^{4} + 13772 T^{5} - 1921862 T^{6} + 82142 T^{7} + 113089342 T^{8} - 43114584 T^{9} - 5653831794 T^{10} + 1443208718 T^{11} + 285781391787 T^{12} + 1443208718 p T^{13} - 5653831794 p^{2} T^{14} - 43114584 p^{3} T^{15} + 113089342 p^{4} T^{16} + 82142 p^{5} T^{17} - 1921862 p^{6} T^{18} + 13772 p^{7} T^{19} + 24796 p^{8} T^{20} - 252 p^{9} T^{21} - 214 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24}
59 (1−13T+5pT2−2839T3+38957T4−294699T5+2963017T6−294699pT7+38957p2T8−2839p3T9+5p5T10−13p5T11+p6T12)2 ( 1 - 13 T + 5 p T^{2} - 2839 T^{3} + 38957 T^{4} - 294699 T^{5} + 2963017 T^{6} - 294699 p T^{7} + 38957 p^{2} T^{8} - 2839 p^{3} T^{9} + 5 p^{5} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )^{2}
61 1−5T−140T2+373T3+8487T4+5202T5−147441T6−963135T7−4711566T8+13690661T9−1296684385T10+689962304T11+162150963097T12+689962304pT13−1296684385p2T14+13690661p3T15−4711566p4T16−963135p5T17−147441p6T18+5202p7T19+8487p8T20+373p9T21−140p10T22−5p11T23+p12T24 1 - 5 T - 140 T^{2} + 373 T^{3} + 8487 T^{4} + 5202 T^{5} - 147441 T^{6} - 963135 T^{7} - 4711566 T^{8} + 13690661 T^{9} - 1296684385 T^{10} + 689962304 T^{11} + 162150963097 T^{12} + 689962304 p T^{13} - 1296684385 p^{2} T^{14} + 13690661 p^{3} T^{15} - 4711566 p^{4} T^{16} - 963135 p^{5} T^{17} - 147441 p^{6} T^{18} + 5202 p^{7} T^{19} + 8487 p^{8} T^{20} + 373 p^{9} T^{21} - 140 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24}
67 1+11T−175T2−2336T3+15663T4+247450T5−15954pT6−18125445T7+60512732T8+977936543T9−2490157221T10−26393757979T11+95373451231T12−26393757979pT13−2490157221p2T14+977936543p3T15+60512732p4T16−18125445p5T17−15954p7T18+247450p7T19+15663p8T20−2336p9T21−175p10T22+11p11T23+p12T24 1 + 11 T - 175 T^{2} - 2336 T^{3} + 15663 T^{4} + 247450 T^{5} - 15954 p T^{6} - 18125445 T^{7} + 60512732 T^{8} + 977936543 T^{9} - 2490157221 T^{10} - 26393757979 T^{11} + 95373451231 T^{12} - 26393757979 p T^{13} - 2490157221 p^{2} T^{14} + 977936543 p^{3} T^{15} + 60512732 p^{4} T^{16} - 18125445 p^{5} T^{17} - 15954 p^{7} T^{18} + 247450 p^{7} T^{19} + 15663 p^{8} T^{20} - 2336 p^{9} T^{21} - 175 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24}
71 1−6T−249T2+278T3+39793T4+68141T5−3761552T6−15648583T7+241594531T8+1275513473T9−10122739162T10−44683203723T11+523547364015T12−44683203723pT13−10122739162p2T14+1275513473p3T15+241594531p4T16−15648583p5T17−3761552p6T18+68141p7T19+39793p8T20+278p9T21−249p10T22−6p11T23+p12T24 1 - 6 T - 249 T^{2} + 278 T^{3} + 39793 T^{4} + 68141 T^{5} - 3761552 T^{6} - 15648583 T^{7} + 241594531 T^{8} + 1275513473 T^{9} - 10122739162 T^{10} - 44683203723 T^{11} + 523547364015 T^{12} - 44683203723 p T^{13} - 10122739162 p^{2} T^{14} + 1275513473 p^{3} T^{15} + 241594531 p^{4} T^{16} - 15648583 p^{5} T^{17} - 3761552 p^{6} T^{18} + 68141 p^{7} T^{19} + 39793 p^{8} T^{20} + 278 p^{9} T^{21} - 249 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24}
73 1−30T+224T2+1118T3−5021T4−290169T5+1854677T6+9817892T7−27971653T8−688598777T9−1819010273T10+20701972840T11+235631264151T12+20701972840pT13−1819010273p2T14−688598777p3T15−27971653p4T16+9817892p5T17+1854677p6T18−290169p7T19−5021p8T20+1118p9T21+224p10T22−30p11T23+p12T24 1 - 30 T + 224 T^{2} + 1118 T^{3} - 5021 T^{4} - 290169 T^{5} + 1854677 T^{6} + 9817892 T^{7} - 27971653 T^{8} - 688598777 T^{9} - 1819010273 T^{10} + 20701972840 T^{11} + 235631264151 T^{12} + 20701972840 p T^{13} - 1819010273 p^{2} T^{14} - 688598777 p^{3} T^{15} - 27971653 p^{4} T^{16} + 9817892 p^{5} T^{17} + 1854677 p^{6} T^{18} - 290169 p^{7} T^{19} - 5021 p^{8} T^{20} + 1118 p^{9} T^{21} + 224 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24}
79 1−7T−277T2+2628T3+34995T4−387429T5−3070086T6+26237658T7+339376855T8−565746882T9−45365142063T10−8895648284T11+4474615429807T12−8895648284pT13−45365142063p2T14−565746882p3T15+339376855p4T16+26237658p5T17−3070086p6T18−387429p7T19+34995p8T20+2628p9T21−277p10T22−7p11T23+p12T24 1 - 7 T - 277 T^{2} + 2628 T^{3} + 34995 T^{4} - 387429 T^{5} - 3070086 T^{6} + 26237658 T^{7} + 339376855 T^{8} - 565746882 T^{9} - 45365142063 T^{10} - 8895648284 T^{11} + 4474615429807 T^{12} - 8895648284 p T^{13} - 45365142063 p^{2} T^{14} - 565746882 p^{3} T^{15} + 339376855 p^{4} T^{16} + 26237658 p^{5} T^{17} - 3070086 p^{6} T^{18} - 387429 p^{7} T^{19} + 34995 p^{8} T^{20} + 2628 p^{9} T^{21} - 277 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24}
83 (1−27T+656T2−10802T3+153994T4−1760871T5+17670883T6−1760871pT7+153994p2T8−10802p3T9+656p4T10−27p5T11+p6T12)2 ( 1 - 27 T + 656 T^{2} - 10802 T^{3} + 153994 T^{4} - 1760871 T^{5} + 17670883 T^{6} - 1760871 p T^{7} + 153994 p^{2} T^{8} - 10802 p^{3} T^{9} + 656 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} )^{2}
89 (1−4T+167T2−1648T3+21035T4−202110T5+2204075T6−202110pT7+21035p2T8−1648p3T9+167p4T10−4p5T11+p6T12)2 ( 1 - 4 T + 167 T^{2} - 1648 T^{3} + 21035 T^{4} - 202110 T^{5} + 2204075 T^{6} - 202110 p T^{7} + 21035 p^{2} T^{8} - 1648 p^{3} T^{9} + 167 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2}
97 1−35T+278T2+3177T3−20496T4−1111333T5+13328183T6+54713297T7−1182920923T8−11775176076T9+251445486222T10−186060844192T11−17274836413101T12−186060844192pT13+251445486222p2T14−11775176076p3T15−1182920923p4T16+54713297p5T17+13328183p6T18−1111333p7T19−20496p8T20+3177p9T21+278p10T22−35p11T23+p12T24 1 - 35 T + 278 T^{2} + 3177 T^{3} - 20496 T^{4} - 1111333 T^{5} + 13328183 T^{6} + 54713297 T^{7} - 1182920923 T^{8} - 11775176076 T^{9} + 251445486222 T^{10} - 186060844192 T^{11} - 17274836413101 T^{12} - 186060844192 p T^{13} + 251445486222 p^{2} T^{14} - 11775176076 p^{3} T^{15} - 1182920923 p^{4} T^{16} + 54713297 p^{5} T^{17} + 13328183 p^{6} T^{18} - 1111333 p^{7} T^{19} - 20496 p^{8} T^{20} + 3177 p^{9} T^{21} + 278 p^{10} T^{22} - 35 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

−3.52683143997779211880704356583, −3.50589051999421539977626693469, −3.45383237494748981207863400597, −3.10619359074108145553187176376, −3.00386571948342543341490619828, −2.92934278500421617752703630122, −2.87563965479997819645837348140, −2.83753304758176781872754146496, −2.52479101997313739219425835019, −2.28768504223823744189439885375, −2.28492842216554993704442385648, −2.19552020756986417624906489263, −2.18762863127060956220944978898, −1.93368373237992002653196236689, −1.66812778624682861465806820869, −1.51937051905495275059087021785, −1.35379884565889016145538646438, −1.34070592054714039825533346688, −1.13233025484344767286346958763, −1.06037455747608530093793971085, −0.909927814578637257170161400004, −0.811997872974530704723468565031, −0.67884128888582850562125857081, −0.57832258904084789940288647068, −0.19818429921661631477491650500, 0.19818429921661631477491650500, 0.57832258904084789940288647068, 0.67884128888582850562125857081, 0.811997872974530704723468565031, 0.909927814578637257170161400004, 1.06037455747608530093793971085, 1.13233025484344767286346958763, 1.34070592054714039825533346688, 1.35379884565889016145538646438, 1.51937051905495275059087021785, 1.66812778624682861465806820869, 1.93368373237992002653196236689, 2.18762863127060956220944978898, 2.19552020756986417624906489263, 2.28492842216554993704442385648, 2.28768504223823744189439885375, 2.52479101997313739219425835019, 2.83753304758176781872754146496, 2.87563965479997819645837348140, 2.92934278500421617752703630122, 3.00386571948342543341490619828, 3.10619359074108145553187176376, 3.45383237494748981207863400597, 3.50589051999421539977626693469, 3.52683143997779211880704356583

Graph of the ZZ-function along the critical line

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