Properties

Label 24-21e12-1.1-c8e12-0-0
Degree $24$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $1.53677\times 10^{11}$
Root an. cond. $2.92488$
Motivic weight $8$
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  + 3·2-s + 486·3-s + 214·4-s + 1.38e3·5-s + 1.45e3·6-s − 1.22e3·7-s + 2.33e3·8-s + 1.24e5·9-s + 4.16e3·10-s + 1.80e4·11-s + 1.04e5·12-s − 3.67e3·14-s + 6.75e5·15-s + 2.47e4·16-s + 6.33e4·17-s + 3.73e5·18-s + 4.76e5·19-s + 2.97e5·20-s − 5.95e5·21-s + 5.42e4·22-s − 6.63e4·23-s + 1.13e6·24-s + 1.88e5·25-s + 2.23e7·27-s − 2.62e5·28-s + 6.71e5·29-s + 2.02e6·30-s + ⋯
L(s)  = 1  + 3/16·2-s + 6·3-s + 0.835·4-s + 2.22·5-s + 9/8·6-s − 0.510·7-s + 0.569·8-s + 19·9-s + 0.416·10-s + 1.23·11-s + 5.01·12-s − 0.0957·14-s + 13.3·15-s + 0.377·16-s + 0.757·17-s + 3.56·18-s + 3.65·19-s + 1.85·20-s − 3.06·21-s + 0.231·22-s − 0.237·23-s + 3.41·24-s + 0.483·25-s + 42·27-s − 0.426·28-s + 0.948·29-s + 2.50·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+4)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.53677\times 10^{11}\)
Root analytic conductor: \(2.92488\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 7^{12} ,\ ( \ : [4]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(221.5977962\)
\(L(\frac12)\) \(\approx\) \(221.5977962\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - p^{4} T + p^{7} T^{2} )^{6} \)
7 \( 1 + 1226 T - 72067 p^{2} T^{2} - 624436674 p^{2} T^{3} - 21803621942 p^{4} T^{4} + 173878498174 p^{7} T^{5} + 1763539547557 p^{10} T^{6} + 173878498174 p^{15} T^{7} - 21803621942 p^{20} T^{8} - 624436674 p^{26} T^{9} - 72067 p^{34} T^{10} + 1226 p^{40} T^{11} + p^{48} T^{12} \)
good2 \( 1 - 3 T - 205 T^{2} - 537 p T^{3} + 7331 p^{2} T^{4} + 213 p^{4} T^{5} - 163095 p^{7} T^{6} + 421593 p^{7} T^{7} + 4789483 p^{8} T^{8} - 8517891 p^{11} T^{9} - 62211029 p^{12} T^{10} + 349968801 p^{14} T^{11} + 24661035361 p^{14} T^{12} + 349968801 p^{22} T^{13} - 62211029 p^{28} T^{14} - 8517891 p^{35} T^{15} + 4789483 p^{40} T^{16} + 421593 p^{47} T^{17} - 163095 p^{55} T^{18} + 213 p^{60} T^{19} + 7331 p^{66} T^{20} - 537 p^{73} T^{21} - 205 p^{80} T^{22} - 3 p^{88} T^{23} + p^{96} T^{24} \)
5 \( 1 - 1389 T + 1740348 T^{2} - 1524067749 T^{3} + 216063252378 p T^{4} - 715155666913833 T^{5} + 386726264924113958 T^{6} - 31639922985843985131 p T^{7} + \)\(10\!\cdots\!94\)\( p^{2} T^{8} + \)\(81\!\cdots\!09\)\( p^{4} T^{9} - \)\(10\!\cdots\!04\)\( p^{4} T^{10} + \)\(36\!\cdots\!81\)\( p^{6} T^{11} - \)\(26\!\cdots\!86\)\( p^{6} T^{12} + \)\(36\!\cdots\!81\)\( p^{14} T^{13} - \)\(10\!\cdots\!04\)\( p^{20} T^{14} + \)\(81\!\cdots\!09\)\( p^{28} T^{15} + \)\(10\!\cdots\!94\)\( p^{34} T^{16} - 31639922985843985131 p^{41} T^{17} + 386726264924113958 p^{48} T^{18} - 715155666913833 p^{56} T^{19} + 216063252378 p^{65} T^{20} - 1524067749 p^{72} T^{21} + 1740348 p^{80} T^{22} - 1389 p^{88} T^{23} + p^{96} T^{24} \)
11 \( 1 - 18081 T - 434956018 T^{2} + 12013583755893 T^{3} + 43985800111924580 T^{4} - \)\(30\!\cdots\!17\)\( T^{5} + \)\(18\!\cdots\!46\)\( T^{6} + \)\(12\!\cdots\!87\)\( p T^{7} - \)\(42\!\cdots\!08\)\( T^{8} + \)\(10\!\cdots\!83\)\( T^{9} - \)\(85\!\cdots\!54\)\( T^{10} - \)\(15\!\cdots\!99\)\( T^{11} + \)\(45\!\cdots\!34\)\( T^{12} - \)\(15\!\cdots\!99\)\( p^{8} T^{13} - \)\(85\!\cdots\!54\)\( p^{16} T^{14} + \)\(10\!\cdots\!83\)\( p^{24} T^{15} - \)\(42\!\cdots\!08\)\( p^{32} T^{16} + \)\(12\!\cdots\!87\)\( p^{41} T^{17} + \)\(18\!\cdots\!46\)\( p^{48} T^{18} - \)\(30\!\cdots\!17\)\( p^{56} T^{19} + 43985800111924580 p^{64} T^{20} + 12013583755893 p^{72} T^{21} - 434956018 p^{80} T^{22} - 18081 p^{88} T^{23} + p^{96} T^{24} \)
13 \( 1 - 4401080643 T^{2} + 8317927976095047747 T^{4} - \)\(85\!\cdots\!80\)\( T^{6} + \)\(49\!\cdots\!29\)\( T^{8} - \)\(12\!\cdots\!25\)\( T^{10} + \)\(16\!\cdots\!06\)\( T^{12} - \)\(12\!\cdots\!25\)\( p^{16} T^{14} + \)\(49\!\cdots\!29\)\( p^{32} T^{16} - \)\(85\!\cdots\!80\)\( p^{48} T^{18} + 8317927976095047747 p^{64} T^{20} - 4401080643 p^{80} T^{22} + p^{96} T^{24} \)
17 \( 1 - 63306 T + 36519735726 T^{2} - 2227348967251284 T^{3} + \)\(72\!\cdots\!01\)\( T^{4} - \)\(46\!\cdots\!24\)\( T^{5} + \)\(61\!\cdots\!66\)\( p T^{6} - \)\(66\!\cdots\!38\)\( T^{7} + \)\(11\!\cdots\!86\)\( T^{8} - \)\(72\!\cdots\!78\)\( T^{9} + \)\(10\!\cdots\!14\)\( T^{10} - \)\(62\!\cdots\!40\)\( T^{11} + \)\(83\!\cdots\!21\)\( T^{12} - \)\(62\!\cdots\!40\)\( p^{8} T^{13} + \)\(10\!\cdots\!14\)\( p^{16} T^{14} - \)\(72\!\cdots\!78\)\( p^{24} T^{15} + \)\(11\!\cdots\!86\)\( p^{32} T^{16} - \)\(66\!\cdots\!38\)\( p^{40} T^{17} + \)\(61\!\cdots\!66\)\( p^{49} T^{18} - \)\(46\!\cdots\!24\)\( p^{56} T^{19} + \)\(72\!\cdots\!01\)\( p^{64} T^{20} - 2227348967251284 p^{72} T^{21} + 36519735726 p^{80} T^{22} - 63306 p^{88} T^{23} + p^{96} T^{24} \)
19 \( 1 - 476265 T + 187374798360 T^{2} - 53229923600955525 T^{3} + \)\(13\!\cdots\!18\)\( T^{4} - \)\(29\!\cdots\!37\)\( T^{5} + \)\(30\!\cdots\!74\)\( p T^{6} - \)\(10\!\cdots\!95\)\( T^{7} + \)\(18\!\cdots\!26\)\( T^{8} - \)\(29\!\cdots\!71\)\( T^{9} + \)\(44\!\cdots\!60\)\( T^{10} - \)\(17\!\cdots\!91\)\( p^{2} T^{11} + \)\(64\!\cdots\!06\)\( p^{4} T^{12} - \)\(17\!\cdots\!91\)\( p^{10} T^{13} + \)\(44\!\cdots\!60\)\( p^{16} T^{14} - \)\(29\!\cdots\!71\)\( p^{24} T^{15} + \)\(18\!\cdots\!26\)\( p^{32} T^{16} - \)\(10\!\cdots\!95\)\( p^{40} T^{17} + \)\(30\!\cdots\!74\)\( p^{49} T^{18} - \)\(29\!\cdots\!37\)\( p^{56} T^{19} + \)\(13\!\cdots\!18\)\( p^{64} T^{20} - 53229923600955525 p^{72} T^{21} + 187374798360 p^{80} T^{22} - 476265 p^{88} T^{23} + p^{96} T^{24} \)
23 \( 1 + 66384 T - 162539325478 T^{2} - 1306288621046592 T^{3} + \)\(42\!\cdots\!69\)\( T^{4} - \)\(98\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!66\)\( T^{6} + \)\(59\!\cdots\!24\)\( T^{7} + \)\(65\!\cdots\!74\)\( T^{8} + \)\(17\!\cdots\!88\)\( T^{9} - \)\(52\!\cdots\!42\)\( T^{10} - \)\(98\!\cdots\!40\)\( T^{11} + \)\(11\!\cdots\!33\)\( T^{12} - \)\(98\!\cdots\!40\)\( p^{8} T^{13} - \)\(52\!\cdots\!42\)\( p^{16} T^{14} + \)\(17\!\cdots\!88\)\( p^{24} T^{15} + \)\(65\!\cdots\!74\)\( p^{32} T^{16} + \)\(59\!\cdots\!24\)\( p^{40} T^{17} + \)\(12\!\cdots\!66\)\( p^{48} T^{18} - \)\(98\!\cdots\!40\)\( p^{56} T^{19} + \)\(42\!\cdots\!69\)\( p^{64} T^{20} - 1306288621046592 p^{72} T^{21} - 162539325478 p^{80} T^{22} + 66384 p^{88} T^{23} + p^{96} T^{24} \)
29 \( ( 1 - 335553 T + 1018769857945 T^{2} - 363322911429581856 T^{3} + \)\(29\!\cdots\!55\)\( p T^{4} - \)\(28\!\cdots\!35\)\( T^{5} + \)\(50\!\cdots\!82\)\( T^{6} - \)\(28\!\cdots\!35\)\( p^{8} T^{7} + \)\(29\!\cdots\!55\)\( p^{17} T^{8} - 363322911429581856 p^{24} T^{9} + 1018769857945 p^{32} T^{10} - 335553 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
31 \( 1 - 717240 T + 3610182194589 T^{2} - 2466376383583206360 T^{3} + \)\(64\!\cdots\!19\)\( T^{4} - \)\(49\!\cdots\!52\)\( T^{5} + \)\(91\!\cdots\!96\)\( T^{6} - \)\(76\!\cdots\!88\)\( T^{7} + \)\(11\!\cdots\!21\)\( T^{8} - \)\(90\!\cdots\!28\)\( T^{9} + \)\(12\!\cdots\!71\)\( T^{10} - \)\(89\!\cdots\!68\)\( T^{11} + \)\(11\!\cdots\!78\)\( T^{12} - \)\(89\!\cdots\!68\)\( p^{8} T^{13} + \)\(12\!\cdots\!71\)\( p^{16} T^{14} - \)\(90\!\cdots\!28\)\( p^{24} T^{15} + \)\(11\!\cdots\!21\)\( p^{32} T^{16} - \)\(76\!\cdots\!88\)\( p^{40} T^{17} + \)\(91\!\cdots\!96\)\( p^{48} T^{18} - \)\(49\!\cdots\!52\)\( p^{56} T^{19} + \)\(64\!\cdots\!19\)\( p^{64} T^{20} - 2466376383583206360 p^{72} T^{21} + 3610182194589 p^{80} T^{22} - 717240 p^{88} T^{23} + p^{96} T^{24} \)
37 \( 1 - 2289443 T - 3892589589502 T^{2} + 24930464740262894627 T^{3} - \)\(25\!\cdots\!80\)\( T^{4} - \)\(62\!\cdots\!99\)\( T^{5} + \)\(21\!\cdots\!18\)\( T^{6} - \)\(16\!\cdots\!81\)\( T^{7} - \)\(39\!\cdots\!12\)\( T^{8} + \)\(10\!\cdots\!97\)\( T^{9} - \)\(44\!\cdots\!26\)\( T^{10} - \)\(13\!\cdots\!97\)\( T^{11} + \)\(37\!\cdots\!30\)\( T^{12} - \)\(13\!\cdots\!97\)\( p^{8} T^{13} - \)\(44\!\cdots\!26\)\( p^{16} T^{14} + \)\(10\!\cdots\!97\)\( p^{24} T^{15} - \)\(39\!\cdots\!12\)\( p^{32} T^{16} - \)\(16\!\cdots\!81\)\( p^{40} T^{17} + \)\(21\!\cdots\!18\)\( p^{48} T^{18} - \)\(62\!\cdots\!99\)\( p^{56} T^{19} - \)\(25\!\cdots\!80\)\( p^{64} T^{20} + 24930464740262894627 p^{72} T^{21} - 3892589589502 p^{80} T^{22} - 2289443 p^{88} T^{23} + p^{96} T^{24} \)
41 \( 1 - 52739269507620 T^{2} + \)\(14\!\cdots\!58\)\( T^{4} - \)\(26\!\cdots\!80\)\( T^{6} + \)\(37\!\cdots\!87\)\( T^{8} - \)\(40\!\cdots\!68\)\( T^{10} + \)\(35\!\cdots\!20\)\( T^{12} - \)\(40\!\cdots\!68\)\( p^{16} T^{14} + \)\(37\!\cdots\!87\)\( p^{32} T^{16} - \)\(26\!\cdots\!80\)\( p^{48} T^{18} + \)\(14\!\cdots\!58\)\( p^{64} T^{20} - 52739269507620 p^{80} T^{22} + p^{96} T^{24} \)
43 \( ( 1 + 5059979 T + 43751415454691 T^{2} + \)\(11\!\cdots\!32\)\( T^{3} + \)\(63\!\cdots\!85\)\( T^{4} + \)\(84\!\cdots\!49\)\( T^{5} + \)\(65\!\cdots\!86\)\( T^{6} + \)\(84\!\cdots\!49\)\( p^{8} T^{7} + \)\(63\!\cdots\!85\)\( p^{16} T^{8} + \)\(11\!\cdots\!32\)\( p^{24} T^{9} + 43751415454691 p^{32} T^{10} + 5059979 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
47 \( 1 + 4198782 T + 116620052275734 T^{2} + \)\(46\!\cdots\!32\)\( T^{3} + \)\(70\!\cdots\!37\)\( T^{4} + \)\(24\!\cdots\!80\)\( T^{5} + \)\(30\!\cdots\!38\)\( T^{6} + \)\(90\!\cdots\!34\)\( T^{7} + \)\(10\!\cdots\!94\)\( T^{8} + \)\(28\!\cdots\!02\)\( T^{9} + \)\(31\!\cdots\!50\)\( T^{10} + \)\(78\!\cdots\!32\)\( T^{11} + \)\(79\!\cdots\!17\)\( T^{12} + \)\(78\!\cdots\!32\)\( p^{8} T^{13} + \)\(31\!\cdots\!50\)\( p^{16} T^{14} + \)\(28\!\cdots\!02\)\( p^{24} T^{15} + \)\(10\!\cdots\!94\)\( p^{32} T^{16} + \)\(90\!\cdots\!34\)\( p^{40} T^{17} + \)\(30\!\cdots\!38\)\( p^{48} T^{18} + \)\(24\!\cdots\!80\)\( p^{56} T^{19} + \)\(70\!\cdots\!37\)\( p^{64} T^{20} + \)\(46\!\cdots\!32\)\( p^{72} T^{21} + 116620052275734 p^{80} T^{22} + 4198782 p^{88} T^{23} + p^{96} T^{24} \)
53 \( 1 + 209511 T - 152791373062720 T^{2} - \)\(13\!\cdots\!65\)\( T^{3} + \)\(26\!\cdots\!50\)\( p T^{4} + \)\(21\!\cdots\!55\)\( T^{5} - \)\(13\!\cdots\!54\)\( T^{6} - \)\(21\!\cdots\!75\)\( T^{7} - \)\(90\!\cdots\!06\)\( T^{8} + \)\(12\!\cdots\!65\)\( T^{9} + \)\(13\!\cdots\!60\)\( T^{10} - \)\(35\!\cdots\!35\)\( T^{11} - \)\(99\!\cdots\!74\)\( T^{12} - \)\(35\!\cdots\!35\)\( p^{8} T^{13} + \)\(13\!\cdots\!60\)\( p^{16} T^{14} + \)\(12\!\cdots\!65\)\( p^{24} T^{15} - \)\(90\!\cdots\!06\)\( p^{32} T^{16} - \)\(21\!\cdots\!75\)\( p^{40} T^{17} - \)\(13\!\cdots\!54\)\( p^{48} T^{18} + \)\(21\!\cdots\!55\)\( p^{56} T^{19} + \)\(26\!\cdots\!50\)\( p^{65} T^{20} - \)\(13\!\cdots\!65\)\( p^{72} T^{21} - 152791373062720 p^{80} T^{22} + 209511 p^{88} T^{23} + p^{96} T^{24} \)
59 \( 1 + 97052259 T + 5155258629473754 T^{2} + \)\(19\!\cdots\!93\)\( T^{3} + \)\(59\!\cdots\!52\)\( T^{4} + \)\(14\!\cdots\!19\)\( T^{5} + \)\(33\!\cdots\!74\)\( T^{6} + \)\(65\!\cdots\!41\)\( T^{7} + \)\(11\!\cdots\!80\)\( T^{8} + \)\(18\!\cdots\!39\)\( T^{9} + \)\(27\!\cdots\!10\)\( T^{10} + \)\(37\!\cdots\!05\)\( T^{11} + \)\(47\!\cdots\!86\)\( T^{12} + \)\(37\!\cdots\!05\)\( p^{8} T^{13} + \)\(27\!\cdots\!10\)\( p^{16} T^{14} + \)\(18\!\cdots\!39\)\( p^{24} T^{15} + \)\(11\!\cdots\!80\)\( p^{32} T^{16} + \)\(65\!\cdots\!41\)\( p^{40} T^{17} + \)\(33\!\cdots\!74\)\( p^{48} T^{18} + \)\(14\!\cdots\!19\)\( p^{56} T^{19} + \)\(59\!\cdots\!52\)\( p^{64} T^{20} + \)\(19\!\cdots\!93\)\( p^{72} T^{21} + 5155258629473754 p^{80} T^{22} + 97052259 p^{88} T^{23} + p^{96} T^{24} \)
61 \( 1 + 24195864 T + 992095881213846 T^{2} + \)\(19\!\cdots\!96\)\( T^{3} + \)\(48\!\cdots\!85\)\( T^{4} + \)\(90\!\cdots\!72\)\( T^{5} + \)\(17\!\cdots\!82\)\( T^{6} + \)\(31\!\cdots\!36\)\( T^{7} + \)\(51\!\cdots\!18\)\( T^{8} + \)\(82\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!06\)\( T^{10} + \)\(18\!\cdots\!20\)\( T^{11} + \)\(25\!\cdots\!29\)\( T^{12} + \)\(18\!\cdots\!20\)\( p^{8} T^{13} + \)\(12\!\cdots\!06\)\( p^{16} T^{14} + \)\(82\!\cdots\!00\)\( p^{24} T^{15} + \)\(51\!\cdots\!18\)\( p^{32} T^{16} + \)\(31\!\cdots\!36\)\( p^{40} T^{17} + \)\(17\!\cdots\!82\)\( p^{48} T^{18} + \)\(90\!\cdots\!72\)\( p^{56} T^{19} + \)\(48\!\cdots\!85\)\( p^{64} T^{20} + \)\(19\!\cdots\!96\)\( p^{72} T^{21} + 992095881213846 p^{80} T^{22} + 24195864 p^{88} T^{23} + p^{96} T^{24} \)
67 \( 1 + 57546057 T + 503043541773180 T^{2} - \)\(87\!\cdots\!07\)\( T^{3} + \)\(66\!\cdots\!42\)\( T^{4} + \)\(22\!\cdots\!53\)\( T^{5} + \)\(13\!\cdots\!86\)\( T^{6} - \)\(77\!\cdots\!93\)\( T^{7} + \)\(66\!\cdots\!78\)\( T^{8} + \)\(56\!\cdots\!35\)\( T^{9} + \)\(20\!\cdots\!44\)\( T^{10} - \)\(44\!\cdots\!41\)\( T^{11} - \)\(18\!\cdots\!90\)\( T^{12} - \)\(44\!\cdots\!41\)\( p^{8} T^{13} + \)\(20\!\cdots\!44\)\( p^{16} T^{14} + \)\(56\!\cdots\!35\)\( p^{24} T^{15} + \)\(66\!\cdots\!78\)\( p^{32} T^{16} - \)\(77\!\cdots\!93\)\( p^{40} T^{17} + \)\(13\!\cdots\!86\)\( p^{48} T^{18} + \)\(22\!\cdots\!53\)\( p^{56} T^{19} + \)\(66\!\cdots\!42\)\( p^{64} T^{20} - \)\(87\!\cdots\!07\)\( p^{72} T^{21} + 503043541773180 p^{80} T^{22} + 57546057 p^{88} T^{23} + p^{96} T^{24} \)
71 \( ( 1 - 112283310 T + 8536915136999002 T^{2} - \)\(44\!\cdots\!34\)\( T^{3} + \)\(18\!\cdots\!11\)\( T^{4} - \)\(63\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!08\)\( T^{6} - \)\(63\!\cdots\!16\)\( p^{8} T^{7} + \)\(18\!\cdots\!11\)\( p^{16} T^{8} - \)\(44\!\cdots\!34\)\( p^{24} T^{9} + 8536915136999002 p^{32} T^{10} - 112283310 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
73 \( 1 + 193344135 T + 20924362777544694 T^{2} + \)\(16\!\cdots\!65\)\( T^{3} + \)\(10\!\cdots\!96\)\( T^{4} + \)\(54\!\cdots\!47\)\( T^{5} + \)\(24\!\cdots\!46\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{7} + \)\(39\!\cdots\!00\)\( T^{8} + \)\(13\!\cdots\!39\)\( T^{9} + \)\(44\!\cdots\!74\)\( T^{10} + \)\(13\!\cdots\!41\)\( T^{11} + \)\(40\!\cdots\!78\)\( T^{12} + \)\(13\!\cdots\!41\)\( p^{8} T^{13} + \)\(44\!\cdots\!74\)\( p^{16} T^{14} + \)\(13\!\cdots\!39\)\( p^{24} T^{15} + \)\(39\!\cdots\!00\)\( p^{32} T^{16} + \)\(10\!\cdots\!61\)\( p^{40} T^{17} + \)\(24\!\cdots\!46\)\( p^{48} T^{18} + \)\(54\!\cdots\!47\)\( p^{56} T^{19} + \)\(10\!\cdots\!96\)\( p^{64} T^{20} + \)\(16\!\cdots\!65\)\( p^{72} T^{21} + 20924362777544694 p^{80} T^{22} + 193344135 p^{88} T^{23} + p^{96} T^{24} \)
79 \( 1 + 29314488 T - 5585169943187355 T^{2} - \)\(12\!\cdots\!24\)\( T^{3} + \)\(17\!\cdots\!35\)\( T^{4} + \)\(26\!\cdots\!00\)\( T^{5} - \)\(40\!\cdots\!96\)\( T^{6} - \)\(44\!\cdots\!12\)\( T^{7} + \)\(77\!\cdots\!01\)\( T^{8} + \)\(58\!\cdots\!56\)\( T^{9} - \)\(13\!\cdots\!25\)\( T^{10} - \)\(35\!\cdots\!08\)\( T^{11} + \)\(20\!\cdots\!22\)\( T^{12} - \)\(35\!\cdots\!08\)\( p^{8} T^{13} - \)\(13\!\cdots\!25\)\( p^{16} T^{14} + \)\(58\!\cdots\!56\)\( p^{24} T^{15} + \)\(77\!\cdots\!01\)\( p^{32} T^{16} - \)\(44\!\cdots\!12\)\( p^{40} T^{17} - \)\(40\!\cdots\!96\)\( p^{48} T^{18} + \)\(26\!\cdots\!00\)\( p^{56} T^{19} + \)\(17\!\cdots\!35\)\( p^{64} T^{20} - \)\(12\!\cdots\!24\)\( p^{72} T^{21} - 5585169943187355 p^{80} T^{22} + 29314488 p^{88} T^{23} + p^{96} T^{24} \)
83 \( 1 - 3685713535725327 T^{2} + \)\(24\!\cdots\!63\)\( T^{4} - \)\(67\!\cdots\!72\)\( T^{6} + \)\(25\!\cdots\!53\)\( T^{8} - \)\(58\!\cdots\!81\)\( T^{10} + \)\(16\!\cdots\!82\)\( T^{12} - \)\(58\!\cdots\!81\)\( p^{16} T^{14} + \)\(25\!\cdots\!53\)\( p^{32} T^{16} - \)\(67\!\cdots\!72\)\( p^{48} T^{18} + \)\(24\!\cdots\!63\)\( p^{64} T^{20} - 3685713535725327 p^{80} T^{22} + p^{96} T^{24} \)
89 \( 1 - 119863098 T + 12917724120211086 T^{2} - \)\(97\!\cdots\!64\)\( T^{3} + \)\(49\!\cdots\!77\)\( T^{4} - \)\(38\!\cdots\!28\)\( T^{5} - \)\(22\!\cdots\!34\)\( T^{6} + \)\(24\!\cdots\!90\)\( T^{7} - \)\(16\!\cdots\!26\)\( T^{8} + \)\(56\!\cdots\!26\)\( T^{9} + \)\(14\!\cdots\!78\)\( T^{10} - \)\(39\!\cdots\!52\)\( T^{11} + \)\(30\!\cdots\!53\)\( T^{12} - \)\(39\!\cdots\!52\)\( p^{8} T^{13} + \)\(14\!\cdots\!78\)\( p^{16} T^{14} + \)\(56\!\cdots\!26\)\( p^{24} T^{15} - \)\(16\!\cdots\!26\)\( p^{32} T^{16} + \)\(24\!\cdots\!90\)\( p^{40} T^{17} - \)\(22\!\cdots\!34\)\( p^{48} T^{18} - \)\(38\!\cdots\!28\)\( p^{56} T^{19} + \)\(49\!\cdots\!77\)\( p^{64} T^{20} - \)\(97\!\cdots\!64\)\( p^{72} T^{21} + 12917724120211086 p^{80} T^{22} - 119863098 p^{88} T^{23} + p^{96} T^{24} \)
97 \( 1 - 52391774941042107 T^{2} + \)\(12\!\cdots\!47\)\( T^{4} - \)\(17\!\cdots\!72\)\( T^{6} + \)\(14\!\cdots\!97\)\( T^{8} - \)\(85\!\cdots\!89\)\( T^{10} + \)\(50\!\cdots\!70\)\( T^{12} - \)\(85\!\cdots\!89\)\( p^{16} T^{14} + \)\(14\!\cdots\!97\)\( p^{32} T^{16} - \)\(17\!\cdots\!72\)\( p^{48} T^{18} + \)\(12\!\cdots\!47\)\( p^{64} T^{20} - 52391774941042107 p^{80} T^{22} + p^{96} T^{24} \)
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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

−4.90451984577133264199041394537, −4.74670573332781030707676964579, −4.40033372889625938006431509199, −4.38736658027689976762218467031, −4.33366838292423950662706734366, −4.22506360976340980350837176937, −3.81450736635306179279823142556, −3.40753390495801416549861755263, −3.23374643559085451961684023377, −3.20668289767648616046254720556, −3.19218146574058159928426590787, −3.17952572251982675574973689158, −3.03059504234991800259725148308, −2.78425934190061825550457288907, −2.46688656610988894716838436107, −2.19538073644178011141474391662, −2.07818564801305892094284391600, −1.99562848237549083504920837924, −1.65927664671602808410626134671, −1.54699572602940162253268319882, −1.35548897443531895879920673947, −1.30359795458164007050240176522, −1.11553333923284367405342848756, −0.76206338658525279302640389267, −0.15601092395109691256074730193, 0.15601092395109691256074730193, 0.76206338658525279302640389267, 1.11553333923284367405342848756, 1.30359795458164007050240176522, 1.35548897443531895879920673947, 1.54699572602940162253268319882, 1.65927664671602808410626134671, 1.99562848237549083504920837924, 2.07818564801305892094284391600, 2.19538073644178011141474391662, 2.46688656610988894716838436107, 2.78425934190061825550457288907, 3.03059504234991800259725148308, 3.17952572251982675574973689158, 3.19218146574058159928426590787, 3.20668289767648616046254720556, 3.23374643559085451961684023377, 3.40753390495801416549861755263, 3.81450736635306179279823142556, 4.22506360976340980350837176937, 4.33366838292423950662706734366, 4.38736658027689976762218467031, 4.40033372889625938006431509199, 4.74670573332781030707676964579, 4.90451984577133264199041394537

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

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