Properties

Degree $24$
Conductor $9.066\times 10^{18}$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s + 21·7-s − 16·8-s + 27·9-s − 9·11-s + 39·13-s + 69·17-s − 462·19-s − 189·21-s − 66·23-s + 144·24-s + 171·25-s − 18·27-s + 159·29-s + 72·31-s + 81·33-s + 1.11e3·37-s − 351·39-s + 147·41-s − 117·43-s + 783·47-s + 1.95e3·49-s − 621·51-s − 249·53-s − 336·56-s + 4.15e3·57-s − 4.24e3·59-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.13·7-s − 0.707·8-s + 9-s − 0.246·11-s + 0.832·13-s + 0.984·17-s − 5.57·19-s − 1.96·21-s − 0.598·23-s + 1.22·24-s + 1.36·25-s − 0.128·27-s + 1.01·29-s + 0.417·31-s + 0.427·33-s + 4.95·37-s − 1.44·39-s + 0.559·41-s − 0.414·43-s + 2.43·47-s + 5.70·49-s − 1.70·51-s − 0.645·53-s − 0.801·56-s + 9.66·57-s − 9.37·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{12} \cdot 19^{12}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{38} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{12} \cdot 19^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.66097\)
\(L(\frac12)\) \(\approx\) \(1.66097\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{3} T^{3} + p^{6} T^{6} )^{2} \)
19 \( 1 + 462 T + 119769 T^{2} + 1138067 p T^{3} + 8295507 p^{2} T^{4} + 48456009 p^{3} T^{5} + 232420374 p^{4} T^{6} + 48456009 p^{6} T^{7} + 8295507 p^{8} T^{8} + 1138067 p^{10} T^{9} + 119769 p^{12} T^{10} + 462 p^{15} T^{11} + p^{18} T^{12} \)
good3 \( 1 + p^{2} T + 2 p^{3} T^{2} + 29 p^{2} T^{3} + 7 p^{4} T^{4} - 44 p^{2} T^{5} - 4724 T^{6} + 647 p^{2} T^{7} + 18802 p^{3} T^{8} + 267433 p^{2} T^{9} - 16606 p^{4} T^{10} - 8562682 p^{2} T^{11} - 485342900 T^{12} - 8562682 p^{5} T^{13} - 16606 p^{10} T^{14} + 267433 p^{11} T^{15} + 18802 p^{15} T^{16} + 647 p^{17} T^{17} - 4724 p^{18} T^{18} - 44 p^{23} T^{19} + 7 p^{28} T^{20} + 29 p^{29} T^{21} + 2 p^{33} T^{22} + p^{35} T^{23} + p^{36} T^{24} \)
5 \( 1 - 171 T^{2} - 432 T^{3} + 684 T^{4} + 189297 T^{5} + 2035721 T^{6} - 20411757 T^{7} - 39405582 T^{8} - 505727712 p T^{9} - 942467184 p T^{10} + 2343884688 p^{3} T^{11} - 629941358634 T^{12} + 2343884688 p^{6} T^{13} - 942467184 p^{7} T^{14} - 505727712 p^{10} T^{15} - 39405582 p^{12} T^{16} - 20411757 p^{15} T^{17} + 2035721 p^{18} T^{18} + 189297 p^{21} T^{19} + 684 p^{24} T^{20} - 432 p^{27} T^{21} - 171 p^{30} T^{22} + p^{36} T^{24} \)
7 \( 1 - 3 p T - 1515 T^{2} + 3540 p T^{3} + 1512579 T^{4} - 17880063 T^{5} - 1075565731 T^{6} + 8262030114 T^{7} + 606007378278 T^{8} - 2582097624714 T^{9} - 275006022177042 T^{10} + 351692577138024 T^{11} + 103612362769606380 T^{12} + 351692577138024 p^{3} T^{13} - 275006022177042 p^{6} T^{14} - 2582097624714 p^{9} T^{15} + 606007378278 p^{12} T^{16} + 8262030114 p^{15} T^{17} - 1075565731 p^{18} T^{18} - 17880063 p^{21} T^{19} + 1512579 p^{24} T^{20} + 3540 p^{28} T^{21} - 1515 p^{30} T^{22} - 3 p^{34} T^{23} + p^{36} T^{24} \)
11 \( 1 + 9 T - 3519 T^{2} + 24496 T^{3} + 5114331 T^{4} - 110124705 T^{5} - 2271860931 T^{6} + 52499450874 T^{7} + 731386484418 T^{8} + 207758908815234 T^{9} - 13017969765518466 T^{10} - 204189988973228292 T^{11} + 30443947856053596940 T^{12} - 204189988973228292 p^{3} T^{13} - 13017969765518466 p^{6} T^{14} + 207758908815234 p^{9} T^{15} + 731386484418 p^{12} T^{16} + 52499450874 p^{15} T^{17} - 2271860931 p^{18} T^{18} - 110124705 p^{21} T^{19} + 5114331 p^{24} T^{20} + 24496 p^{27} T^{21} - 3519 p^{30} T^{22} + 9 p^{33} T^{23} + p^{36} T^{24} \)
13 \( 1 - 3 p T + 3093 T^{2} - 98440 T^{3} + 11349087 T^{4} - 378701979 T^{5} + 22862720507 T^{6} - 1116317647566 T^{7} + 56385479661678 T^{8} - 3687796202409564 T^{9} + 159033549430554174 T^{10} - 10099287187718310156 T^{11} + \)\(30\!\cdots\!88\)\( T^{12} - 10099287187718310156 p^{3} T^{13} + 159033549430554174 p^{6} T^{14} - 3687796202409564 p^{9} T^{15} + 56385479661678 p^{12} T^{16} - 1116317647566 p^{15} T^{17} + 22862720507 p^{18} T^{18} - 378701979 p^{21} T^{19} + 11349087 p^{24} T^{20} - 98440 p^{27} T^{21} + 3093 p^{30} T^{22} - 3 p^{34} T^{23} + p^{36} T^{24} \)
17 \( 1 - 69 T + 6573 T^{2} + 298204 T^{3} - 55154319 T^{4} + 270492363 p T^{5} - 148240032933 T^{6} - 16787667391872 T^{7} + 1892272804048104 T^{8} - 125048706972526476 T^{9} + 913900858267910370 T^{10} + 33791091237853613706 p T^{11} - \)\(17\!\cdots\!28\)\( p^{2} T^{12} + 33791091237853613706 p^{4} T^{13} + 913900858267910370 p^{6} T^{14} - 125048706972526476 p^{9} T^{15} + 1892272804048104 p^{12} T^{16} - 16787667391872 p^{15} T^{17} - 148240032933 p^{18} T^{18} + 270492363 p^{22} T^{19} - 55154319 p^{24} T^{20} + 298204 p^{27} T^{21} + 6573 p^{30} T^{22} - 69 p^{33} T^{23} + p^{36} T^{24} \)
23 \( 1 + 66 T - 9393 T^{2} + 671695 T^{3} + 134377509 T^{4} - 14874284241 T^{5} + 71997450171 T^{6} + 263342907791031 T^{7} - 15774109169407374 T^{8} - 4876569355971214692 T^{9} + \)\(29\!\cdots\!14\)\( T^{10} + \)\(83\!\cdots\!47\)\( T^{11} - \)\(84\!\cdots\!52\)\( T^{12} + \)\(83\!\cdots\!47\)\( p^{3} T^{13} + \)\(29\!\cdots\!14\)\( p^{6} T^{14} - 4876569355971214692 p^{9} T^{15} - 15774109169407374 p^{12} T^{16} + 263342907791031 p^{15} T^{17} + 71997450171 p^{18} T^{18} - 14874284241 p^{21} T^{19} + 134377509 p^{24} T^{20} + 671695 p^{27} T^{21} - 9393 p^{30} T^{22} + 66 p^{33} T^{23} + p^{36} T^{24} \)
29 \( 1 - 159 T + 36549 T^{2} - 3331656 T^{3} + 421841088 T^{4} - 3272692197 p T^{5} + 5535241134320 T^{6} + 625908025808775 T^{7} + 1040540252412762 p T^{8} - 48469236719521320171 T^{9} + \)\(14\!\cdots\!33\)\( T^{10} - \)\(23\!\cdots\!52\)\( T^{11} + \)\(17\!\cdots\!06\)\( T^{12} - \)\(23\!\cdots\!52\)\( p^{3} T^{13} + \)\(14\!\cdots\!33\)\( p^{6} T^{14} - 48469236719521320171 p^{9} T^{15} + 1040540252412762 p^{13} T^{16} + 625908025808775 p^{15} T^{17} + 5535241134320 p^{18} T^{18} - 3272692197 p^{22} T^{19} + 421841088 p^{24} T^{20} - 3331656 p^{27} T^{21} + 36549 p^{30} T^{22} - 159 p^{33} T^{23} + p^{36} T^{24} \)
31 \( 1 - 72 T - 84360 T^{2} - 4099822 T^{3} + 4645966785 T^{4} + 582123802836 T^{5} - 97979947426741 T^{6} - 42244835246634435 T^{7} - 818163153401404386 T^{8} + \)\(13\!\cdots\!80\)\( T^{9} + \)\(66\!\cdots\!43\)\( p T^{10} - \)\(20\!\cdots\!07\)\( T^{11} - \)\(78\!\cdots\!40\)\( T^{12} - \)\(20\!\cdots\!07\)\( p^{3} T^{13} + \)\(66\!\cdots\!43\)\( p^{7} T^{14} + \)\(13\!\cdots\!80\)\( p^{9} T^{15} - 818163153401404386 p^{12} T^{16} - 42244835246634435 p^{15} T^{17} - 97979947426741 p^{18} T^{18} + 582123802836 p^{21} T^{19} + 4645966785 p^{24} T^{20} - 4099822 p^{27} T^{21} - 84360 p^{30} T^{22} - 72 p^{33} T^{23} + p^{36} T^{24} \)
37 \( ( 1 - 558 T + 298089 T^{2} - 113545809 T^{3} + 38178310329 T^{4} - 10429273769733 T^{5} + 69667380457282 p T^{6} - 10429273769733 p^{3} T^{7} + 38178310329 p^{6} T^{8} - 113545809 p^{9} T^{9} + 298089 p^{12} T^{10} - 558 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
41 \( 1 - 147 T + 44007 T^{2} + 21769246 T^{3} - 2350025733 T^{4} + 234418847265 T^{5} + 457706944154289 T^{6} - 9303997209190392 T^{7} - 6140319093154185000 T^{8} + \)\(10\!\cdots\!34\)\( T^{9} - \)\(12\!\cdots\!74\)\( T^{10} - \)\(45\!\cdots\!46\)\( T^{11} + \)\(16\!\cdots\!04\)\( T^{12} - \)\(45\!\cdots\!46\)\( p^{3} T^{13} - \)\(12\!\cdots\!74\)\( p^{6} T^{14} + \)\(10\!\cdots\!34\)\( p^{9} T^{15} - 6140319093154185000 p^{12} T^{16} - 9303997209190392 p^{15} T^{17} + 457706944154289 p^{18} T^{18} + 234418847265 p^{21} T^{19} - 2350025733 p^{24} T^{20} + 21769246 p^{27} T^{21} + 44007 p^{30} T^{22} - 147 p^{33} T^{23} + p^{36} T^{24} \)
43 \( 1 + 117 T - 100119 T^{2} + 38915762 T^{3} + 2994100635 T^{4} - 3671912050095 T^{5} + 1310121343946135 T^{6} - 70375225814813832 T^{7} - 74777917746500954580 T^{8} + \)\(41\!\cdots\!76\)\( T^{9} - \)\(21\!\cdots\!80\)\( T^{10} - \)\(11\!\cdots\!68\)\( T^{11} + \)\(88\!\cdots\!32\)\( T^{12} - \)\(11\!\cdots\!68\)\( p^{3} T^{13} - \)\(21\!\cdots\!80\)\( p^{6} T^{14} + \)\(41\!\cdots\!76\)\( p^{9} T^{15} - 74777917746500954580 p^{12} T^{16} - 70375225814813832 p^{15} T^{17} + 1310121343946135 p^{18} T^{18} - 3671912050095 p^{21} T^{19} + 2994100635 p^{24} T^{20} + 38915762 p^{27} T^{21} - 100119 p^{30} T^{22} + 117 p^{33} T^{23} + p^{36} T^{24} \)
47 \( 1 - 783 T + 538857 T^{2} - 254728868 T^{3} + 112205777895 T^{4} - 37288296975249 T^{5} + 10875425952712257 T^{6} - 1895616375918277806 T^{7} + 46254528927632446254 T^{8} + \)\(26\!\cdots\!66\)\( T^{9} - \)\(31\!\cdots\!62\)\( p T^{10} + \)\(68\!\cdots\!52\)\( T^{11} - \)\(22\!\cdots\!72\)\( T^{12} + \)\(68\!\cdots\!52\)\( p^{3} T^{13} - \)\(31\!\cdots\!62\)\( p^{7} T^{14} + \)\(26\!\cdots\!66\)\( p^{9} T^{15} + 46254528927632446254 p^{12} T^{16} - 1895616375918277806 p^{15} T^{17} + 10875425952712257 p^{18} T^{18} - 37288296975249 p^{21} T^{19} + 112205777895 p^{24} T^{20} - 254728868 p^{27} T^{21} + 538857 p^{30} T^{22} - 783 p^{33} T^{23} + p^{36} T^{24} \)
53 \( 1 + 249 T + 105357 T^{2} + 53055920 T^{3} + 56278965819 T^{4} + 10000173124845 T^{5} + 3156327827474091 T^{6} + 2244063380209899702 T^{7} + \)\(94\!\cdots\!94\)\( T^{8} + \)\(24\!\cdots\!32\)\( T^{9} + \)\(31\!\cdots\!86\)\( T^{10} + \)\(47\!\cdots\!40\)\( T^{11} + \)\(98\!\cdots\!48\)\( T^{12} + \)\(47\!\cdots\!40\)\( p^{3} T^{13} + \)\(31\!\cdots\!86\)\( p^{6} T^{14} + \)\(24\!\cdots\!32\)\( p^{9} T^{15} + \)\(94\!\cdots\!94\)\( p^{12} T^{16} + 2244063380209899702 p^{15} T^{17} + 3156327827474091 p^{18} T^{18} + 10000173124845 p^{21} T^{19} + 56278965819 p^{24} T^{20} + 53055920 p^{27} T^{21} + 105357 p^{30} T^{22} + 249 p^{33} T^{23} + p^{36} T^{24} \)
59 \( 1 + 72 p T + 147897 p T^{2} + 11515262701 T^{3} + 10887706714419 T^{4} + 7727213299360905 T^{5} + 4135284193453945767 T^{6} + \)\(15\!\cdots\!07\)\( T^{7} + \)\(29\!\cdots\!98\)\( T^{8} - \)\(11\!\cdots\!96\)\( T^{9} - \)\(15\!\cdots\!02\)\( T^{10} - \)\(10\!\cdots\!07\)\( T^{11} - \)\(50\!\cdots\!20\)\( T^{12} - \)\(10\!\cdots\!07\)\( p^{3} T^{13} - \)\(15\!\cdots\!02\)\( p^{6} T^{14} - \)\(11\!\cdots\!96\)\( p^{9} T^{15} + \)\(29\!\cdots\!98\)\( p^{12} T^{16} + \)\(15\!\cdots\!07\)\( p^{15} T^{17} + 4135284193453945767 p^{18} T^{18} + 7727213299360905 p^{21} T^{19} + 10887706714419 p^{24} T^{20} + 11515262701 p^{27} T^{21} + 147897 p^{31} T^{22} + 72 p^{34} T^{23} + p^{36} T^{24} \)
61 \( 1 - 3114 T + 4298859 T^{2} - 3204919049 T^{3} + 998006352417 T^{4} + 482660655435063 T^{5} - 719435635956944909 T^{6} + \)\(34\!\cdots\!33\)\( T^{7} - \)\(35\!\cdots\!32\)\( T^{8} - \)\(50\!\cdots\!02\)\( T^{9} + \)\(27\!\cdots\!08\)\( T^{10} - \)\(17\!\cdots\!79\)\( T^{11} - \)\(25\!\cdots\!48\)\( T^{12} - \)\(17\!\cdots\!79\)\( p^{3} T^{13} + \)\(27\!\cdots\!08\)\( p^{6} T^{14} - \)\(50\!\cdots\!02\)\( p^{9} T^{15} - \)\(35\!\cdots\!32\)\( p^{12} T^{16} + \)\(34\!\cdots\!33\)\( p^{15} T^{17} - 719435635956944909 p^{18} T^{18} + 482660655435063 p^{21} T^{19} + 998006352417 p^{24} T^{20} - 3204919049 p^{27} T^{21} + 4298859 p^{30} T^{22} - 3114 p^{33} T^{23} + p^{36} T^{24} \)
67 \( 1 - 3060 T + 4343541 T^{2} - 3539177707 T^{3} + 1674046544745 T^{4} - 364838652101685 T^{5} - 33128397443652253 T^{6} + 59458849334550877089 T^{7} - \)\(73\!\cdots\!64\)\( T^{8} + \)\(81\!\cdots\!52\)\( T^{9} - \)\(50\!\cdots\!24\)\( T^{10} + \)\(16\!\cdots\!97\)\( T^{11} - \)\(43\!\cdots\!88\)\( T^{12} + \)\(16\!\cdots\!97\)\( p^{3} T^{13} - \)\(50\!\cdots\!24\)\( p^{6} T^{14} + \)\(81\!\cdots\!52\)\( p^{9} T^{15} - \)\(73\!\cdots\!64\)\( p^{12} T^{16} + 59458849334550877089 p^{15} T^{17} - 33128397443652253 p^{18} T^{18} - 364838652101685 p^{21} T^{19} + 1674046544745 p^{24} T^{20} - 3539177707 p^{27} T^{21} + 4343541 p^{30} T^{22} - 3060 p^{33} T^{23} + p^{36} T^{24} \)
71 \( 1 - 1686 T + 18357 p T^{2} - 332046979 T^{3} - 395700826839 T^{4} + 496154621895921 T^{5} - 192091304753971689 T^{6} - \)\(10\!\cdots\!95\)\( T^{7} + \)\(20\!\cdots\!46\)\( T^{8} - \)\(13\!\cdots\!76\)\( T^{9} + \)\(27\!\cdots\!34\)\( T^{10} + \)\(26\!\cdots\!85\)\( T^{11} - \)\(28\!\cdots\!24\)\( T^{12} + \)\(26\!\cdots\!85\)\( p^{3} T^{13} + \)\(27\!\cdots\!34\)\( p^{6} T^{14} - \)\(13\!\cdots\!76\)\( p^{9} T^{15} + \)\(20\!\cdots\!46\)\( p^{12} T^{16} - \)\(10\!\cdots\!95\)\( p^{15} T^{17} - 192091304753971689 p^{18} T^{18} + 496154621895921 p^{21} T^{19} - 395700826839 p^{24} T^{20} - 332046979 p^{27} T^{21} + 18357 p^{31} T^{22} - 1686 p^{33} T^{23} + p^{36} T^{24} \)
73 \( 1 - 1626 T + 1342599 T^{2} - 77826027 T^{3} - 482426009259 T^{4} + 337597036991529 T^{5} + 152387349969643313 T^{6} - \)\(21\!\cdots\!53\)\( T^{7} + \)\(80\!\cdots\!70\)\( T^{8} + \)\(61\!\cdots\!48\)\( T^{9} - \)\(37\!\cdots\!06\)\( T^{10} - \)\(65\!\cdots\!35\)\( T^{11} + \)\(17\!\cdots\!44\)\( T^{12} - \)\(65\!\cdots\!35\)\( p^{3} T^{13} - \)\(37\!\cdots\!06\)\( p^{6} T^{14} + \)\(61\!\cdots\!48\)\( p^{9} T^{15} + \)\(80\!\cdots\!70\)\( p^{12} T^{16} - \)\(21\!\cdots\!53\)\( p^{15} T^{17} + 152387349969643313 p^{18} T^{18} + 337597036991529 p^{21} T^{19} - 482426009259 p^{24} T^{20} - 77826027 p^{27} T^{21} + 1342599 p^{30} T^{22} - 1626 p^{33} T^{23} + p^{36} T^{24} \)
79 \( 1 + 327 T - 352623 T^{2} + 85160886 T^{3} - 74868578124 T^{4} - 561682606508301 T^{5} - 10832178836590636 T^{6} + \)\(15\!\cdots\!35\)\( T^{7} - \)\(13\!\cdots\!24\)\( T^{8} + \)\(11\!\cdots\!49\)\( T^{9} + \)\(69\!\cdots\!71\)\( T^{10} - \)\(11\!\cdots\!46\)\( T^{11} - \)\(30\!\cdots\!66\)\( T^{12} - \)\(11\!\cdots\!46\)\( p^{3} T^{13} + \)\(69\!\cdots\!71\)\( p^{6} T^{14} + \)\(11\!\cdots\!49\)\( p^{9} T^{15} - \)\(13\!\cdots\!24\)\( p^{12} T^{16} + \)\(15\!\cdots\!35\)\( p^{15} T^{17} - 10832178836590636 p^{18} T^{18} - 561682606508301 p^{21} T^{19} - 74868578124 p^{24} T^{20} + 85160886 p^{27} T^{21} - 352623 p^{30} T^{22} + 327 p^{33} T^{23} + p^{36} T^{24} \)
83 \( 1 - 927 T - 1910169 T^{2} + 1162039358 T^{3} + 2637336171501 T^{4} - 771225603730341 T^{5} - 2584240489804536525 T^{6} + \)\(27\!\cdots\!66\)\( T^{7} + \)\(19\!\cdots\!86\)\( T^{8} - \)\(17\!\cdots\!40\)\( T^{9} - \)\(12\!\cdots\!02\)\( T^{10} - \)\(85\!\cdots\!76\)\( T^{11} + \)\(72\!\cdots\!04\)\( T^{12} - \)\(85\!\cdots\!76\)\( p^{3} T^{13} - \)\(12\!\cdots\!02\)\( p^{6} T^{14} - \)\(17\!\cdots\!40\)\( p^{9} T^{15} + \)\(19\!\cdots\!86\)\( p^{12} T^{16} + \)\(27\!\cdots\!66\)\( p^{15} T^{17} - 2584240489804536525 p^{18} T^{18} - 771225603730341 p^{21} T^{19} + 2637336171501 p^{24} T^{20} + 1162039358 p^{27} T^{21} - 1910169 p^{30} T^{22} - 927 p^{33} T^{23} + p^{36} T^{24} \)
89 \( 1 + 6366 T + 18939837 T^{2} + 33490901356 T^{3} + 36794607216366 T^{4} + 23062320055435131 T^{5} + 6331029765277091109 T^{6} + \)\(65\!\cdots\!65\)\( T^{7} + \)\(26\!\cdots\!56\)\( p T^{8} + \)\(34\!\cdots\!44\)\( T^{9} + \)\(26\!\cdots\!92\)\( T^{10} + \)\(82\!\cdots\!30\)\( T^{11} + \)\(98\!\cdots\!30\)\( T^{12} + \)\(82\!\cdots\!30\)\( p^{3} T^{13} + \)\(26\!\cdots\!92\)\( p^{6} T^{14} + \)\(34\!\cdots\!44\)\( p^{9} T^{15} + \)\(26\!\cdots\!56\)\( p^{13} T^{16} + \)\(65\!\cdots\!65\)\( p^{15} T^{17} + 6331029765277091109 p^{18} T^{18} + 23062320055435131 p^{21} T^{19} + 36794607216366 p^{24} T^{20} + 33490901356 p^{27} T^{21} + 18939837 p^{30} T^{22} + 6366 p^{33} T^{23} + p^{36} T^{24} \)
97 \( 1 + 8052 T + 31462761 T^{2} + 80299670389 T^{3} + 153146011957221 T^{4} + 239405956107270819 T^{5} + \)\(33\!\cdots\!19\)\( T^{6} + \)\(42\!\cdots\!83\)\( T^{7} + \)\(52\!\cdots\!34\)\( T^{8} + \)\(60\!\cdots\!74\)\( T^{9} + \)\(65\!\cdots\!46\)\( T^{10} + \)\(66\!\cdots\!83\)\( T^{11} + \)\(64\!\cdots\!84\)\( T^{12} + \)\(66\!\cdots\!83\)\( p^{3} T^{13} + \)\(65\!\cdots\!46\)\( p^{6} T^{14} + \)\(60\!\cdots\!74\)\( p^{9} T^{15} + \)\(52\!\cdots\!34\)\( p^{12} T^{16} + \)\(42\!\cdots\!83\)\( p^{15} T^{17} + \)\(33\!\cdots\!19\)\( p^{18} T^{18} + 239405956107270819 p^{21} T^{19} + 153146011957221 p^{24} T^{20} + 80299670389 p^{27} T^{21} + 31462761 p^{30} T^{22} + 8052 p^{33} T^{23} + p^{36} T^{24} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.67016626590436642342248672287, −5.66818470012601980930414069586, −5.50470822912783472397258660228, −5.37238790592604125968192353903, −5.34594045531414258216042501909, −5.24613765911295703480756290836, −4.81679317681798222354397783719, −4.42113063449723259283772279558, −4.38973376250632290109302048616, −4.26053320818227431816275109165, −4.20956562096483613172292710688, −4.20506019753539861250375713101, −4.16375678502870495109587636081, −3.72296144094416059821015388658, −3.63908570122628391781077396456, −2.98592509311963802813407473184, −2.77174712317525807257827391304, −2.61519432144845590195005176877, −2.58992815477760479930580117309, −2.22922113351296215776659011489, −2.15254292565823380774123943343, −1.50395117912655908937421382552, −1.06648730716889019959033399643, −0.72500703526505515992456136430, −0.43792185813842336477761371441, 0.43792185813842336477761371441, 0.72500703526505515992456136430, 1.06648730716889019959033399643, 1.50395117912655908937421382552, 2.15254292565823380774123943343, 2.22922113351296215776659011489, 2.58992815477760479930580117309, 2.61519432144845590195005176877, 2.77174712317525807257827391304, 2.98592509311963802813407473184, 3.63908570122628391781077396456, 3.72296144094416059821015388658, 4.16375678502870495109587636081, 4.20506019753539861250375713101, 4.20956562096483613172292710688, 4.26053320818227431816275109165, 4.38973376250632290109302048616, 4.42113063449723259283772279558, 4.81679317681798222354397783719, 5.24613765911295703480756290836, 5.34594045531414258216042501909, 5.37238790592604125968192353903, 5.50470822912783472397258660228, 5.66818470012601980930414069586, 5.67016626590436642342248672287

Graph of the $Z$-function along the critical line

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