Properties

Label 24-91e12-1.1-c7e12-0-0
Degree $24$
Conductor $3.225\times 10^{23}$
Sign $1$
Analytic cond. $2.78470\times 10^{17}$
Root an. cond. $5.33170$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 82·3-s − 257·4-s + 1.02e3·5-s + 492·6-s + 4.11e3·7-s − 1.79e3·8-s − 4.30e3·9-s + 6.15e3·10-s + 1.21e4·11-s − 2.10e4·12-s + 2.63e4·13-s + 2.46e4·14-s + 8.41e4·15-s + 2.17e4·16-s + 8.27e4·17-s − 2.58e4·18-s − 1.03e4·19-s − 2.63e5·20-s + 3.37e5·21-s + 7.30e4·22-s + 9.83e4·23-s − 1.47e5·24-s + 1.93e5·25-s + 1.58e5·26-s − 4.94e5·27-s − 1.05e6·28-s + ⋯
L(s)  = 1  + 0.530·2-s + 1.75·3-s − 2.00·4-s + 3.67·5-s + 0.929·6-s + 4.53·7-s − 1.23·8-s − 1.97·9-s + 1.94·10-s + 2.75·11-s − 3.52·12-s + 3.32·13-s + 2.40·14-s + 6.43·15-s + 1.32·16-s + 4.08·17-s − 1.04·18-s − 0.344·19-s − 7.37·20-s + 7.95·21-s + 1.46·22-s + 1.68·23-s − 2.17·24-s + 2.48·25-s + 1.76·26-s − 4.83·27-s − 9.10·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(7^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(2.78470\times 10^{17}\)
Root analytic conductor: \(5.33170\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{91} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 7^{12} \cdot 13^{12} ,\ ( \ : [7/2]^{12} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(3162.849829\)
\(L(\frac12)\) \(\approx\) \(3162.849829\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( ( 1 - p^{3} T )^{12} \)
13 \( ( 1 - p^{3} T )^{12} \)
good2 \( 1 - 3 p T + 293 T^{2} - 753 p T^{3} + 25905 p T^{4} - 65397 p^{2} T^{5} + 1689145 p^{2} T^{6} - 1014387 p^{4} T^{7} + 2558775 p^{8} T^{8} + 3583491 p^{7} T^{9} + 1088599741 p^{6} T^{10} + 1007913375 p^{9} T^{11} + 3478719271 p^{11} T^{12} + 1007913375 p^{16} T^{13} + 1088599741 p^{20} T^{14} + 3583491 p^{28} T^{15} + 2558775 p^{36} T^{16} - 1014387 p^{39} T^{17} + 1689145 p^{44} T^{18} - 65397 p^{51} T^{19} + 25905 p^{57} T^{20} - 753 p^{64} T^{21} + 293 p^{70} T^{22} - 3 p^{78} T^{23} + p^{84} T^{24} \)
3 \( 1 - 82 T + 11033 T^{2} - 254452 p T^{3} + 68208946 T^{4} - 4297559948 T^{5} + 300850180049 T^{6} - 5864706681146 p T^{7} + 116632020987847 p^{2} T^{8} - 701796714785204 p^{4} T^{9} + 36946291920388910 p^{4} T^{10} - 616257336045973216 p^{5} T^{11} + 9787752756029642908 p^{6} T^{12} - 616257336045973216 p^{12} T^{13} + 36946291920388910 p^{18} T^{14} - 701796714785204 p^{25} T^{15} + 116632020987847 p^{30} T^{16} - 5864706681146 p^{36} T^{17} + 300850180049 p^{42} T^{18} - 4297559948 p^{49} T^{19} + 68208946 p^{56} T^{20} - 254452 p^{64} T^{21} + 11033 p^{70} T^{22} - 82 p^{77} T^{23} + p^{84} T^{24} \)
5 \( 1 - 1026 T + 171744 p T^{2} - 525923796 T^{3} + 272450446356 T^{4} - 24360695944602 p T^{5} + 9596737496776896 p T^{6} - 686119261168056558 p^{2} T^{7} + 44784683762963988836 p^{3} T^{8} - \)\(27\!\cdots\!96\)\( p^{4} T^{9} + \)\(15\!\cdots\!64\)\( p^{5} T^{10} - \)\(89\!\cdots\!98\)\( p^{6} T^{11} + \)\(50\!\cdots\!26\)\( p^{7} T^{12} - \)\(89\!\cdots\!98\)\( p^{13} T^{13} + \)\(15\!\cdots\!64\)\( p^{19} T^{14} - \)\(27\!\cdots\!96\)\( p^{25} T^{15} + 44784683762963988836 p^{31} T^{16} - 686119261168056558 p^{37} T^{17} + 9596737496776896 p^{43} T^{18} - 24360695944602 p^{50} T^{19} + 272450446356 p^{56} T^{20} - 525923796 p^{63} T^{21} + 171744 p^{71} T^{22} - 1026 p^{77} T^{23} + p^{84} T^{24} \)
11 \( 1 - 12168 T + 192909859 T^{2} - 1805032374234 T^{3} + 17311105740499694 T^{4} - \)\(13\!\cdots\!86\)\( T^{5} + \)\(96\!\cdots\!67\)\( T^{6} - \)\(61\!\cdots\!16\)\( T^{7} + \)\(37\!\cdots\!79\)\( T^{8} - \)\(20\!\cdots\!20\)\( T^{9} + \)\(11\!\cdots\!06\)\( T^{10} - \)\(53\!\cdots\!48\)\( T^{11} + \)\(24\!\cdots\!52\)\( T^{12} - \)\(53\!\cdots\!48\)\( p^{7} T^{13} + \)\(11\!\cdots\!06\)\( p^{14} T^{14} - \)\(20\!\cdots\!20\)\( p^{21} T^{15} + \)\(37\!\cdots\!79\)\( p^{28} T^{16} - \)\(61\!\cdots\!16\)\( p^{35} T^{17} + \)\(96\!\cdots\!67\)\( p^{42} T^{18} - \)\(13\!\cdots\!86\)\( p^{49} T^{19} + 17311105740499694 p^{56} T^{20} - 1805032374234 p^{63} T^{21} + 192909859 p^{70} T^{22} - 12168 p^{77} T^{23} + p^{84} T^{24} \)
17 \( 1 - 82710 T + 5909480354 T^{2} - 281931708666018 T^{3} + 12125522649869541114 T^{4} - \)\(42\!\cdots\!90\)\( T^{5} + \)\(13\!\cdots\!38\)\( T^{6} - \)\(39\!\cdots\!34\)\( T^{7} + \)\(10\!\cdots\!39\)\( T^{8} - \)\(26\!\cdots\!72\)\( T^{9} + \)\(62\!\cdots\!28\)\( T^{10} - \)\(13\!\cdots\!24\)\( T^{11} + \)\(28\!\cdots\!12\)\( T^{12} - \)\(13\!\cdots\!24\)\( p^{7} T^{13} + \)\(62\!\cdots\!28\)\( p^{14} T^{14} - \)\(26\!\cdots\!72\)\( p^{21} T^{15} + \)\(10\!\cdots\!39\)\( p^{28} T^{16} - \)\(39\!\cdots\!34\)\( p^{35} T^{17} + \)\(13\!\cdots\!38\)\( p^{42} T^{18} - \)\(42\!\cdots\!90\)\( p^{49} T^{19} + 12125522649869541114 p^{56} T^{20} - 281931708666018 p^{63} T^{21} + 5909480354 p^{70} T^{22} - 82710 p^{77} T^{23} + p^{84} T^{24} \)
19 \( 1 + 10302 T + 3937982284 T^{2} + 22552674130752 T^{3} + 7502982250033985224 T^{4} - \)\(22\!\cdots\!22\)\( T^{5} + \)\(98\!\cdots\!80\)\( T^{6} - \)\(12\!\cdots\!78\)\( T^{7} + \)\(11\!\cdots\!68\)\( T^{8} - \)\(19\!\cdots\!64\)\( T^{9} + \)\(65\!\cdots\!72\)\( p T^{10} - \)\(19\!\cdots\!34\)\( T^{11} + \)\(12\!\cdots\!30\)\( T^{12} - \)\(19\!\cdots\!34\)\( p^{7} T^{13} + \)\(65\!\cdots\!72\)\( p^{15} T^{14} - \)\(19\!\cdots\!64\)\( p^{21} T^{15} + \)\(11\!\cdots\!68\)\( p^{28} T^{16} - \)\(12\!\cdots\!78\)\( p^{35} T^{17} + \)\(98\!\cdots\!80\)\( p^{42} T^{18} - \)\(22\!\cdots\!22\)\( p^{49} T^{19} + 7502982250033985224 p^{56} T^{20} + 22552674130752 p^{63} T^{21} + 3937982284 p^{70} T^{22} + 10302 p^{77} T^{23} + p^{84} T^{24} \)
23 \( 1 - 98376 T + 22585713559 T^{2} - 1753447816456416 T^{3} + \)\(23\!\cdots\!74\)\( T^{4} - \)\(15\!\cdots\!48\)\( T^{5} + \)\(15\!\cdots\!57\)\( T^{6} - \)\(94\!\cdots\!16\)\( T^{7} + \)\(79\!\cdots\!80\)\( T^{8} - \)\(43\!\cdots\!16\)\( T^{9} + \)\(32\!\cdots\!63\)\( T^{10} - \)\(16\!\cdots\!16\)\( T^{11} + \)\(11\!\cdots\!04\)\( T^{12} - \)\(16\!\cdots\!16\)\( p^{7} T^{13} + \)\(32\!\cdots\!63\)\( p^{14} T^{14} - \)\(43\!\cdots\!16\)\( p^{21} T^{15} + \)\(79\!\cdots\!80\)\( p^{28} T^{16} - \)\(94\!\cdots\!16\)\( p^{35} T^{17} + \)\(15\!\cdots\!57\)\( p^{42} T^{18} - \)\(15\!\cdots\!48\)\( p^{49} T^{19} + \)\(23\!\cdots\!74\)\( p^{56} T^{20} - 1753447816456416 p^{63} T^{21} + 22585713559 p^{70} T^{22} - 98376 p^{77} T^{23} + p^{84} T^{24} \)
29 \( 1 - 350592 T + 155012690392 T^{2} - 45308071052655612 T^{3} + \)\(12\!\cdots\!48\)\( T^{4} - \)\(29\!\cdots\!96\)\( T^{5} + \)\(62\!\cdots\!24\)\( T^{6} - \)\(12\!\cdots\!80\)\( T^{7} + \)\(22\!\cdots\!32\)\( T^{8} - \)\(37\!\cdots\!84\)\( T^{9} + \)\(57\!\cdots\!48\)\( T^{10} - \)\(83\!\cdots\!36\)\( T^{11} + \)\(11\!\cdots\!50\)\( T^{12} - \)\(83\!\cdots\!36\)\( p^{7} T^{13} + \)\(57\!\cdots\!48\)\( p^{14} T^{14} - \)\(37\!\cdots\!84\)\( p^{21} T^{15} + \)\(22\!\cdots\!32\)\( p^{28} T^{16} - \)\(12\!\cdots\!80\)\( p^{35} T^{17} + \)\(62\!\cdots\!24\)\( p^{42} T^{18} - \)\(29\!\cdots\!96\)\( p^{49} T^{19} + \)\(12\!\cdots\!48\)\( p^{56} T^{20} - 45308071052655612 p^{63} T^{21} + 155012690392 p^{70} T^{22} - 350592 p^{77} T^{23} + p^{84} T^{24} \)
31 \( 1 - 55092 T + 140480987049 T^{2} - 5076916577652124 T^{3} + \)\(95\!\cdots\!70\)\( T^{4} - \)\(12\!\cdots\!12\)\( T^{5} + \)\(43\!\cdots\!43\)\( T^{6} + \)\(71\!\cdots\!24\)\( T^{7} + \)\(15\!\cdots\!20\)\( T^{8} + \)\(75\!\cdots\!76\)\( T^{9} + \)\(44\!\cdots\!85\)\( T^{10} + \)\(34\!\cdots\!88\)\( T^{11} + \)\(12\!\cdots\!04\)\( T^{12} + \)\(34\!\cdots\!88\)\( p^{7} T^{13} + \)\(44\!\cdots\!85\)\( p^{14} T^{14} + \)\(75\!\cdots\!76\)\( p^{21} T^{15} + \)\(15\!\cdots\!20\)\( p^{28} T^{16} + \)\(71\!\cdots\!24\)\( p^{35} T^{17} + \)\(43\!\cdots\!43\)\( p^{42} T^{18} - \)\(12\!\cdots\!12\)\( p^{49} T^{19} + \)\(95\!\cdots\!70\)\( p^{56} T^{20} - 5076916577652124 p^{63} T^{21} + 140480987049 p^{70} T^{22} - 55092 p^{77} T^{23} + p^{84} T^{24} \)
37 \( 1 - 376310 T + 686968664645 T^{2} - 200160445852473024 T^{3} + \)\(22\!\cdots\!78\)\( T^{4} - \)\(52\!\cdots\!64\)\( T^{5} + \)\(48\!\cdots\!05\)\( T^{6} - \)\(91\!\cdots\!14\)\( T^{7} + \)\(75\!\cdots\!31\)\( T^{8} - \)\(11\!\cdots\!68\)\( T^{9} + \)\(93\!\cdots\!78\)\( T^{10} - \)\(12\!\cdots\!32\)\( T^{11} + \)\(96\!\cdots\!16\)\( T^{12} - \)\(12\!\cdots\!32\)\( p^{7} T^{13} + \)\(93\!\cdots\!78\)\( p^{14} T^{14} - \)\(11\!\cdots\!68\)\( p^{21} T^{15} + \)\(75\!\cdots\!31\)\( p^{28} T^{16} - \)\(91\!\cdots\!14\)\( p^{35} T^{17} + \)\(48\!\cdots\!05\)\( p^{42} T^{18} - \)\(52\!\cdots\!64\)\( p^{49} T^{19} + \)\(22\!\cdots\!78\)\( p^{56} T^{20} - 200160445852473024 p^{63} T^{21} + 686968664645 p^{70} T^{22} - 376310 p^{77} T^{23} + p^{84} T^{24} \)
41 \( 1 - 1387272 T + 2498146104023 T^{2} - 2432102444465698536 T^{3} + \)\(25\!\cdots\!56\)\( T^{4} - \)\(19\!\cdots\!12\)\( T^{5} + \)\(15\!\cdots\!47\)\( T^{6} - \)\(99\!\cdots\!80\)\( T^{7} + \)\(64\!\cdots\!47\)\( T^{8} - \)\(34\!\cdots\!16\)\( T^{9} + \)\(19\!\cdots\!58\)\( T^{10} - \)\(90\!\cdots\!60\)\( T^{11} + \)\(43\!\cdots\!28\)\( T^{12} - \)\(90\!\cdots\!60\)\( p^{7} T^{13} + \)\(19\!\cdots\!58\)\( p^{14} T^{14} - \)\(34\!\cdots\!16\)\( p^{21} T^{15} + \)\(64\!\cdots\!47\)\( p^{28} T^{16} - \)\(99\!\cdots\!80\)\( p^{35} T^{17} + \)\(15\!\cdots\!47\)\( p^{42} T^{18} - \)\(19\!\cdots\!12\)\( p^{49} T^{19} + \)\(25\!\cdots\!56\)\( p^{56} T^{20} - 2432102444465698536 p^{63} T^{21} + 2498146104023 p^{70} T^{22} - 1387272 p^{77} T^{23} + p^{84} T^{24} \)
43 \( 1 - 568708 T + 1979453955984 T^{2} - 889504972066334528 T^{3} + \)\(18\!\cdots\!40\)\( T^{4} - \)\(65\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} - \)\(30\!\cdots\!76\)\( T^{7} + \)\(45\!\cdots\!36\)\( T^{8} - \)\(10\!\cdots\!16\)\( T^{9} + \)\(15\!\cdots\!88\)\( T^{10} - \)\(30\!\cdots\!60\)\( T^{11} + \)\(45\!\cdots\!06\)\( T^{12} - \)\(30\!\cdots\!60\)\( p^{7} T^{13} + \)\(15\!\cdots\!88\)\( p^{14} T^{14} - \)\(10\!\cdots\!16\)\( p^{21} T^{15} + \)\(45\!\cdots\!36\)\( p^{28} T^{16} - \)\(30\!\cdots\!76\)\( p^{35} T^{17} + \)\(10\!\cdots\!76\)\( p^{42} T^{18} - \)\(65\!\cdots\!68\)\( p^{49} T^{19} + \)\(18\!\cdots\!40\)\( p^{56} T^{20} - 889504972066334528 p^{63} T^{21} + 1979453955984 p^{70} T^{22} - 568708 p^{77} T^{23} + p^{84} T^{24} \)
47 \( 1 - 1359444 T + 4421174546565 T^{2} - 4726813291608700092 T^{3} + \)\(89\!\cdots\!22\)\( T^{4} - \)\(80\!\cdots\!64\)\( T^{5} + \)\(11\!\cdots\!59\)\( T^{6} - \)\(91\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!76\)\( T^{8} - \)\(75\!\cdots\!32\)\( T^{9} + \)\(76\!\cdots\!73\)\( T^{10} - \)\(48\!\cdots\!68\)\( T^{11} + \)\(43\!\cdots\!72\)\( T^{12} - \)\(48\!\cdots\!68\)\( p^{7} T^{13} + \)\(76\!\cdots\!73\)\( p^{14} T^{14} - \)\(75\!\cdots\!32\)\( p^{21} T^{15} + \)\(10\!\cdots\!76\)\( p^{28} T^{16} - \)\(91\!\cdots\!20\)\( p^{35} T^{17} + \)\(11\!\cdots\!59\)\( p^{42} T^{18} - \)\(80\!\cdots\!64\)\( p^{49} T^{19} + \)\(89\!\cdots\!22\)\( p^{56} T^{20} - 4726813291608700092 p^{63} T^{21} + 4421174546565 p^{70} T^{22} - 1359444 p^{77} T^{23} + p^{84} T^{24} \)
53 \( 1 - 2061780 T + 8187253625068 T^{2} - 277226000830950144 p T^{3} + \)\(34\!\cdots\!72\)\( T^{4} - \)\(54\!\cdots\!04\)\( T^{5} + \)\(96\!\cdots\!44\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(20\!\cdots\!20\)\( T^{8} - \)\(25\!\cdots\!48\)\( T^{9} + \)\(62\!\cdots\!08\)\( p T^{10} - \)\(36\!\cdots\!04\)\( T^{11} + \)\(43\!\cdots\!30\)\( T^{12} - \)\(36\!\cdots\!04\)\( p^{7} T^{13} + \)\(62\!\cdots\!08\)\( p^{15} T^{14} - \)\(25\!\cdots\!48\)\( p^{21} T^{15} + \)\(20\!\cdots\!20\)\( p^{28} T^{16} - \)\(13\!\cdots\!20\)\( p^{35} T^{17} + \)\(96\!\cdots\!44\)\( p^{42} T^{18} - \)\(54\!\cdots\!04\)\( p^{49} T^{19} + \)\(34\!\cdots\!72\)\( p^{56} T^{20} - 277226000830950144 p^{64} T^{21} + 8187253625068 p^{70} T^{22} - 2061780 p^{77} T^{23} + p^{84} T^{24} \)
59 \( 1 - 395964 T + 16920750961476 T^{2} - 2654758660435840428 T^{3} + \)\(13\!\cdots\!66\)\( T^{4} + \)\(18\!\cdots\!68\)\( T^{5} + \)\(65\!\cdots\!52\)\( T^{6} + \)\(31\!\cdots\!68\)\( T^{7} + \)\(24\!\cdots\!19\)\( T^{8} + \)\(18\!\cdots\!08\)\( T^{9} + \)\(72\!\cdots\!92\)\( T^{10} + \)\(67\!\cdots\!72\)\( T^{11} + \)\(19\!\cdots\!68\)\( T^{12} + \)\(67\!\cdots\!72\)\( p^{7} T^{13} + \)\(72\!\cdots\!92\)\( p^{14} T^{14} + \)\(18\!\cdots\!08\)\( p^{21} T^{15} + \)\(24\!\cdots\!19\)\( p^{28} T^{16} + \)\(31\!\cdots\!68\)\( p^{35} T^{17} + \)\(65\!\cdots\!52\)\( p^{42} T^{18} + \)\(18\!\cdots\!68\)\( p^{49} T^{19} + \)\(13\!\cdots\!66\)\( p^{56} T^{20} - 2654758660435840428 p^{63} T^{21} + 16920750961476 p^{70} T^{22} - 395964 p^{77} T^{23} + p^{84} T^{24} \)
61 \( 1 - 444006 T + 17262081715805 T^{2} - 7810652070056865598 T^{3} + \)\(16\!\cdots\!52\)\( T^{4} - \)\(62\!\cdots\!30\)\( T^{5} + \)\(11\!\cdots\!05\)\( T^{6} - \)\(35\!\cdots\!86\)\( T^{7} + \)\(57\!\cdots\!59\)\( T^{8} - \)\(16\!\cdots\!96\)\( T^{9} + \)\(24\!\cdots\!14\)\( T^{10} - \)\(61\!\cdots\!52\)\( T^{11} + \)\(83\!\cdots\!04\)\( T^{12} - \)\(61\!\cdots\!52\)\( p^{7} T^{13} + \)\(24\!\cdots\!14\)\( p^{14} T^{14} - \)\(16\!\cdots\!96\)\( p^{21} T^{15} + \)\(57\!\cdots\!59\)\( p^{28} T^{16} - \)\(35\!\cdots\!86\)\( p^{35} T^{17} + \)\(11\!\cdots\!05\)\( p^{42} T^{18} - \)\(62\!\cdots\!30\)\( p^{49} T^{19} + \)\(16\!\cdots\!52\)\( p^{56} T^{20} - 7810652070056865598 p^{63} T^{21} + 17262081715805 p^{70} T^{22} - 444006 p^{77} T^{23} + p^{84} T^{24} \)
67 \( 1 + 3094010 T + 28399424821569 T^{2} + 68266124353984765342 T^{3} + \)\(42\!\cdots\!88\)\( T^{4} + \)\(89\!\cdots\!74\)\( T^{5} + \)\(47\!\cdots\!73\)\( T^{6} + \)\(88\!\cdots\!38\)\( T^{7} + \)\(42\!\cdots\!71\)\( T^{8} + \)\(73\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!90\)\( T^{10} + \)\(51\!\cdots\!20\)\( T^{11} + \)\(21\!\cdots\!04\)\( T^{12} + \)\(51\!\cdots\!20\)\( p^{7} T^{13} + \)\(32\!\cdots\!90\)\( p^{14} T^{14} + \)\(73\!\cdots\!08\)\( p^{21} T^{15} + \)\(42\!\cdots\!71\)\( p^{28} T^{16} + \)\(88\!\cdots\!38\)\( p^{35} T^{17} + \)\(47\!\cdots\!73\)\( p^{42} T^{18} + \)\(89\!\cdots\!74\)\( p^{49} T^{19} + \)\(42\!\cdots\!88\)\( p^{56} T^{20} + 68266124353984765342 p^{63} T^{21} + 28399424821569 p^{70} T^{22} + 3094010 p^{77} T^{23} + p^{84} T^{24} \)
71 \( 1 - 5694366 T + 97691124763942 T^{2} - \)\(46\!\cdots\!10\)\( T^{3} + \)\(61\!\cdots\!38\)\( p T^{4} - \)\(17\!\cdots\!98\)\( T^{5} + \)\(12\!\cdots\!22\)\( T^{6} - \)\(42\!\cdots\!66\)\( T^{7} + \)\(23\!\cdots\!51\)\( T^{8} - \)\(71\!\cdots\!76\)\( T^{9} + \)\(32\!\cdots\!68\)\( T^{10} - \)\(87\!\cdots\!24\)\( T^{11} + \)\(33\!\cdots\!56\)\( T^{12} - \)\(87\!\cdots\!24\)\( p^{7} T^{13} + \)\(32\!\cdots\!68\)\( p^{14} T^{14} - \)\(71\!\cdots\!76\)\( p^{21} T^{15} + \)\(23\!\cdots\!51\)\( p^{28} T^{16} - \)\(42\!\cdots\!66\)\( p^{35} T^{17} + \)\(12\!\cdots\!22\)\( p^{42} T^{18} - \)\(17\!\cdots\!98\)\( p^{49} T^{19} + \)\(61\!\cdots\!38\)\( p^{57} T^{20} - \)\(46\!\cdots\!10\)\( p^{63} T^{21} + 97691124763942 p^{70} T^{22} - 5694366 p^{77} T^{23} + p^{84} T^{24} \)
73 \( 1 - 7052346 T + 93803283536335 T^{2} - \)\(54\!\cdots\!70\)\( T^{3} + \)\(42\!\cdots\!34\)\( T^{4} - \)\(20\!\cdots\!30\)\( T^{5} + \)\(12\!\cdots\!57\)\( T^{6} - \)\(52\!\cdots\!78\)\( T^{7} + \)\(25\!\cdots\!68\)\( T^{8} - \)\(95\!\cdots\!10\)\( T^{9} + \)\(39\!\cdots\!95\)\( T^{10} - \)\(13\!\cdots\!14\)\( T^{11} + \)\(49\!\cdots\!64\)\( T^{12} - \)\(13\!\cdots\!14\)\( p^{7} T^{13} + \)\(39\!\cdots\!95\)\( p^{14} T^{14} - \)\(95\!\cdots\!10\)\( p^{21} T^{15} + \)\(25\!\cdots\!68\)\( p^{28} T^{16} - \)\(52\!\cdots\!78\)\( p^{35} T^{17} + \)\(12\!\cdots\!57\)\( p^{42} T^{18} - \)\(20\!\cdots\!30\)\( p^{49} T^{19} + \)\(42\!\cdots\!34\)\( p^{56} T^{20} - \)\(54\!\cdots\!70\)\( p^{63} T^{21} + 93803283536335 p^{70} T^{22} - 7052346 p^{77} T^{23} + p^{84} T^{24} \)
79 \( 1 - 4304160 T + 122786972103123 T^{2} - \)\(47\!\cdots\!80\)\( T^{3} + \)\(71\!\cdots\!10\)\( T^{4} - \)\(25\!\cdots\!96\)\( T^{5} + \)\(27\!\cdots\!25\)\( T^{6} - \)\(84\!\cdots\!56\)\( T^{7} + \)\(77\!\cdots\!60\)\( T^{8} - \)\(21\!\cdots\!64\)\( T^{9} + \)\(18\!\cdots\!71\)\( T^{10} - \)\(45\!\cdots\!00\)\( T^{11} + \)\(36\!\cdots\!00\)\( T^{12} - \)\(45\!\cdots\!00\)\( p^{7} T^{13} + \)\(18\!\cdots\!71\)\( p^{14} T^{14} - \)\(21\!\cdots\!64\)\( p^{21} T^{15} + \)\(77\!\cdots\!60\)\( p^{28} T^{16} - \)\(84\!\cdots\!56\)\( p^{35} T^{17} + \)\(27\!\cdots\!25\)\( p^{42} T^{18} - \)\(25\!\cdots\!96\)\( p^{49} T^{19} + \)\(71\!\cdots\!10\)\( p^{56} T^{20} - \)\(47\!\cdots\!80\)\( p^{63} T^{21} + 122786972103123 p^{70} T^{22} - 4304160 p^{77} T^{23} + p^{84} T^{24} \)
83 \( 1 - 2704554 T + 132649670810416 T^{2} - \)\(36\!\cdots\!04\)\( T^{3} + \)\(92\!\cdots\!48\)\( T^{4} - \)\(24\!\cdots\!46\)\( T^{5} + \)\(46\!\cdots\!56\)\( T^{6} - \)\(11\!\cdots\!30\)\( T^{7} + \)\(19\!\cdots\!56\)\( T^{8} - \)\(43\!\cdots\!92\)\( T^{9} + \)\(65\!\cdots\!36\)\( T^{10} - \)\(14\!\cdots\!74\)\( T^{11} + \)\(19\!\cdots\!42\)\( T^{12} - \)\(14\!\cdots\!74\)\( p^{7} T^{13} + \)\(65\!\cdots\!36\)\( p^{14} T^{14} - \)\(43\!\cdots\!92\)\( p^{21} T^{15} + \)\(19\!\cdots\!56\)\( p^{28} T^{16} - \)\(11\!\cdots\!30\)\( p^{35} T^{17} + \)\(46\!\cdots\!56\)\( p^{42} T^{18} - \)\(24\!\cdots\!46\)\( p^{49} T^{19} + \)\(92\!\cdots\!48\)\( p^{56} T^{20} - \)\(36\!\cdots\!04\)\( p^{63} T^{21} + 132649670810416 p^{70} T^{22} - 2704554 p^{77} T^{23} + p^{84} T^{24} \)
89 \( 1 + 10986042 T + 348067547296072 T^{2} + \)\(28\!\cdots\!80\)\( T^{3} + \)\(52\!\cdots\!80\)\( T^{4} + \)\(31\!\cdots\!46\)\( T^{5} + \)\(46\!\cdots\!76\)\( T^{6} + \)\(19\!\cdots\!50\)\( T^{7} + \)\(27\!\cdots\!52\)\( T^{8} + \)\(72\!\cdots\!16\)\( T^{9} + \)\(12\!\cdots\!32\)\( T^{10} + \)\(17\!\cdots\!18\)\( T^{11} + \)\(54\!\cdots\!74\)\( T^{12} + \)\(17\!\cdots\!18\)\( p^{7} T^{13} + \)\(12\!\cdots\!32\)\( p^{14} T^{14} + \)\(72\!\cdots\!16\)\( p^{21} T^{15} + \)\(27\!\cdots\!52\)\( p^{28} T^{16} + \)\(19\!\cdots\!50\)\( p^{35} T^{17} + \)\(46\!\cdots\!76\)\( p^{42} T^{18} + \)\(31\!\cdots\!46\)\( p^{49} T^{19} + \)\(52\!\cdots\!80\)\( p^{56} T^{20} + \)\(28\!\cdots\!80\)\( p^{63} T^{21} + 348067547296072 p^{70} T^{22} + 10986042 p^{77} T^{23} + p^{84} T^{24} \)
97 \( 1 + 24462382 T + 995586091407367 T^{2} + \)\(18\!\cdots\!06\)\( T^{3} + \)\(43\!\cdots\!18\)\( T^{4} + \)\(66\!\cdots\!26\)\( T^{5} + \)\(11\!\cdots\!25\)\( T^{6} + \)\(14\!\cdots\!46\)\( T^{7} + \)\(20\!\cdots\!92\)\( T^{8} + \)\(22\!\cdots\!86\)\( T^{9} + \)\(25\!\cdots\!59\)\( T^{10} + \)\(25\!\cdots\!54\)\( p T^{11} + \)\(24\!\cdots\!76\)\( T^{12} + \)\(25\!\cdots\!54\)\( p^{8} T^{13} + \)\(25\!\cdots\!59\)\( p^{14} T^{14} + \)\(22\!\cdots\!86\)\( p^{21} T^{15} + \)\(20\!\cdots\!92\)\( p^{28} T^{16} + \)\(14\!\cdots\!46\)\( p^{35} T^{17} + \)\(11\!\cdots\!25\)\( p^{42} T^{18} + \)\(66\!\cdots\!26\)\( p^{49} T^{19} + \)\(43\!\cdots\!18\)\( p^{56} T^{20} + \)\(18\!\cdots\!06\)\( p^{63} T^{21} + 995586091407367 p^{70} T^{22} + 24462382 p^{77} T^{23} + p^{84} T^{24} \)
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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

−3.49287209667505932660659997569, −3.34718796833597043485414713386, −3.10108183696087487386120079086, −3.08258049307173903683321344673, −3.05642196884090982834010073390, −2.94674567585944454400392899466, −2.78861643359035278617693336246, −2.75749677327559843177147731676, −2.34505854555864858862763852033, −2.23568639822199071509877170261, −2.14066581243329772853750142319, −1.93617478239699795212098804694, −1.90825323555893724232474652023, −1.77821097335086865608385862557, −1.77316910285534149119624601364, −1.46985071615891315727496169394, −1.36877779662944549026067084814, −1.12709111371301620697117808331, −1.08740823339933789418069275016, −0.895882643931233893724177015307, −0.836010255614980589089843058380, −0.73285204090141573717617691639, −0.59501242932623183041398140908, −0.57955221890404332003864515888, −0.47044971842319153896974196468, 0.47044971842319153896974196468, 0.57955221890404332003864515888, 0.59501242932623183041398140908, 0.73285204090141573717617691639, 0.836010255614980589089843058380, 0.895882643931233893724177015307, 1.08740823339933789418069275016, 1.12709111371301620697117808331, 1.36877779662944549026067084814, 1.46985071615891315727496169394, 1.77316910285534149119624601364, 1.77821097335086865608385862557, 1.90825323555893724232474652023, 1.93617478239699795212098804694, 2.14066581243329772853750142319, 2.23568639822199071509877170261, 2.34505854555864858862763852033, 2.75749677327559843177147731676, 2.78861643359035278617693336246, 2.94674567585944454400392899466, 3.05642196884090982834010073390, 3.08258049307173903683321344673, 3.10108183696087487386120079086, 3.34718796833597043485414713386, 3.49287209667505932660659997569

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

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