# 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 of factors

## 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.