Properties

Label 24-405e12-1.1-c3e12-0-0
Degree $24$
Conductor $1.947\times 10^{31}$
Sign $1$
Analytic cond. $3.46619\times 10^{16}$
Root an. cond. $4.88833$
Motivic weight $3$
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  − 4·2-s + 15·4-s − 30·5-s − 40·7-s − 16·8-s + 120·10-s − 88·11-s − 20·13-s + 160·14-s + 54·16-s + 248·17-s − 92·19-s − 450·20-s + 352·22-s − 210·23-s + 375·25-s + 80·26-s − 600·28-s − 296·29-s + 104·31-s − 94·32-s − 992·34-s + 1.20e3·35-s − 408·37-s + 368·38-s + 480·40-s − 344·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 15/8·4-s − 2.68·5-s − 2.15·7-s − 0.707·8-s + 3.79·10-s − 2.41·11-s − 0.426·13-s + 3.05·14-s + 0.843·16-s + 3.53·17-s − 1.11·19-s − 5.03·20-s + 3.41·22-s − 1.90·23-s + 3·25-s + 0.603·26-s − 4.04·28-s − 1.89·29-s + 0.602·31-s − 0.519·32-s − 5.00·34-s + 5.79·35-s − 1.81·37-s + 1.57·38-s + 1.89·40-s − 1.31·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{48} \cdot 5^{12}\)
Sign: $1$
Analytic conductor: \(3.46619\times 10^{16}\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{405} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{48} \cdot 5^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.1947448484\)
\(L(\frac12)\) \(\approx\) \(0.1947448484\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 + p T + p^{2} T^{2} )^{6} \)
good2 \( 1 + p^{2} T + T^{2} - 5 p^{3} T^{3} - 165 T^{4} - 83 p T^{5} + 653 p T^{6} + 1089 p^{2} T^{7} + 29 p^{6} T^{8} - 2365 p^{3} T^{9} - 7833 p^{3} T^{10} + 1571 p^{4} T^{11} + 34673 p^{4} T^{12} + 1571 p^{7} T^{13} - 7833 p^{9} T^{14} - 2365 p^{12} T^{15} + 29 p^{18} T^{16} + 1089 p^{17} T^{17} + 653 p^{19} T^{18} - 83 p^{22} T^{19} - 165 p^{24} T^{20} - 5 p^{30} T^{21} + p^{30} T^{22} + p^{35} T^{23} + p^{36} T^{24} \)
7 \( 1 + 40 T - 195 T^{2} - 27600 T^{3} - 120614 T^{4} + 9594736 T^{5} + 58964223 T^{6} - 2627326272 T^{7} - 15033425480 T^{8} + 839040384384 T^{9} + 1546325002115 p T^{10} - 167540869013400 T^{11} - 6479444384354948 T^{12} - 167540869013400 p^{3} T^{13} + 1546325002115 p^{7} T^{14} + 839040384384 p^{9} T^{15} - 15033425480 p^{12} T^{16} - 2627326272 p^{15} T^{17} + 58964223 p^{18} T^{18} + 9594736 p^{21} T^{19} - 120614 p^{24} T^{20} - 27600 p^{27} T^{21} - 195 p^{30} T^{22} + 40 p^{33} T^{23} + p^{36} T^{24} \)
11 \( 1 + 8 p T - 160 p T^{2} - 325888 T^{3} + 4699968 T^{4} + 993665000 T^{5} + 65416660 T^{6} - 1659313374552 T^{7} - 3369759338368 T^{8} + 2205393212773120 T^{9} + 25018569959095200 T^{10} - 87500462789453240 p T^{11} - 230363066231836282 p^{2} T^{12} - 87500462789453240 p^{4} T^{13} + 25018569959095200 p^{6} T^{14} + 2205393212773120 p^{9} T^{15} - 3369759338368 p^{12} T^{16} - 1659313374552 p^{15} T^{17} + 65416660 p^{18} T^{18} + 993665000 p^{21} T^{19} + 4699968 p^{24} T^{20} - 325888 p^{27} T^{21} - 160 p^{31} T^{22} + 8 p^{34} T^{23} + p^{36} T^{24} \)
13 \( 1 + 20 T - 8263 T^{2} - 6092 p T^{3} + 37778358 T^{4} + 64342332 T^{5} - 112147890233 T^{6} + 731238500156 T^{7} + 245449791222944 T^{8} - 2666824331395180 T^{9} - 422065538614557691 T^{10} + 3139084705670288516 T^{11} + \)\(79\!\cdots\!00\)\( T^{12} + 3139084705670288516 p^{3} T^{13} - 422065538614557691 p^{6} T^{14} - 2666824331395180 p^{9} T^{15} + 245449791222944 p^{12} T^{16} + 731238500156 p^{15} T^{17} - 112147890233 p^{18} T^{18} + 64342332 p^{21} T^{19} + 37778358 p^{24} T^{20} - 6092 p^{28} T^{21} - 8263 p^{30} T^{22} + 20 p^{33} T^{23} + p^{36} T^{24} \)
17 \( ( 1 - 124 T + 24474 T^{2} - 134308 p T^{3} + 259934659 T^{4} - 18722277280 T^{5} + 1609913951204 T^{6} - 18722277280 p^{3} T^{7} + 259934659 p^{6} T^{8} - 134308 p^{10} T^{9} + 24474 p^{12} T^{10} - 124 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
19 \( ( 1 + 46 T + 22985 T^{2} + 705414 T^{3} + 253542922 T^{4} + 5221909598 T^{5} + 1923956438809 T^{6} + 5221909598 p^{3} T^{7} + 253542922 p^{6} T^{8} + 705414 p^{9} T^{9} + 22985 p^{12} T^{10} + 46 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
23 \( 1 + 210 T - 3123 T^{2} - 5016750 T^{3} - 439138206 T^{4} + 37136510850 T^{5} + 6380112231275 T^{6} - 356383291480470 T^{7} - 87871269065796360 T^{8} + 3555770380478971290 T^{9} + \)\(13\!\cdots\!57\)\( T^{10} - \)\(33\!\cdots\!30\)\( T^{11} - \)\(15\!\cdots\!96\)\( T^{12} - \)\(33\!\cdots\!30\)\( p^{3} T^{13} + \)\(13\!\cdots\!57\)\( p^{6} T^{14} + 3555770380478971290 p^{9} T^{15} - 87871269065796360 p^{12} T^{16} - 356383291480470 p^{15} T^{17} + 6380112231275 p^{18} T^{18} + 37136510850 p^{21} T^{19} - 439138206 p^{24} T^{20} - 5016750 p^{27} T^{21} - 3123 p^{30} T^{22} + 210 p^{33} T^{23} + p^{36} T^{24} \)
29 \( 1 + 296 T - 69344 T^{2} - 19179488 T^{3} + 5168997408 T^{4} + 907838101096 T^{5} - 269271769187756 T^{6} - 28603253623039080 T^{7} + 11175888646383523424 T^{8} + \)\(62\!\cdots\!92\)\( T^{9} - \)\(36\!\cdots\!04\)\( T^{10} - \)\(55\!\cdots\!92\)\( T^{11} + \)\(10\!\cdots\!46\)\( T^{12} - \)\(55\!\cdots\!92\)\( p^{3} T^{13} - \)\(36\!\cdots\!04\)\( p^{6} T^{14} + \)\(62\!\cdots\!92\)\( p^{9} T^{15} + 11175888646383523424 p^{12} T^{16} - 28603253623039080 p^{15} T^{17} - 269271769187756 p^{18} T^{18} + 907838101096 p^{21} T^{19} + 5168997408 p^{24} T^{20} - 19179488 p^{27} T^{21} - 69344 p^{30} T^{22} + 296 p^{33} T^{23} + p^{36} T^{24} \)
31 \( 1 - 104 T - 109656 T^{2} + 15695928 T^{3} + 6167871064 T^{4} - 1068076508720 T^{5} - 214938559690308 T^{6} + 45673654323044928 T^{7} + 4910634484218985960 T^{8} - \)\(12\!\cdots\!48\)\( T^{9} - \)\(72\!\cdots\!36\)\( T^{10} + \)\(14\!\cdots\!36\)\( T^{11} + \)\(11\!\cdots\!86\)\( T^{12} + \)\(14\!\cdots\!36\)\( p^{3} T^{13} - \)\(72\!\cdots\!36\)\( p^{6} T^{14} - \)\(12\!\cdots\!48\)\( p^{9} T^{15} + 4910634484218985960 p^{12} T^{16} + 45673654323044928 p^{15} T^{17} - 214938559690308 p^{18} T^{18} - 1068076508720 p^{21} T^{19} + 6167871064 p^{24} T^{20} + 15695928 p^{27} T^{21} - 109656 p^{30} T^{22} - 104 p^{33} T^{23} + p^{36} T^{24} \)
37 \( ( 1 + 204 T + 229518 T^{2} + 31409012 T^{3} + 23106132267 T^{4} + 2366964443472 T^{5} + 1434960824277324 T^{6} + 2366964443472 p^{3} T^{7} + 23106132267 p^{6} T^{8} + 31409012 p^{9} T^{9} + 229518 p^{12} T^{10} + 204 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
41 \( 1 + 344 T - 109997 T^{2} - 62132888 T^{3} - 2568517965 T^{4} + 2673285031072 T^{5} + 575954726100244 T^{6} + 94369533842064624 T^{7} + 16845556915121356121 T^{8} - \)\(11\!\cdots\!76\)\( T^{9} - \)\(54\!\cdots\!67\)\( T^{10} + \)\(27\!\cdots\!72\)\( T^{11} + \)\(47\!\cdots\!26\)\( T^{12} + \)\(27\!\cdots\!72\)\( p^{3} T^{13} - \)\(54\!\cdots\!67\)\( p^{6} T^{14} - \)\(11\!\cdots\!76\)\( p^{9} T^{15} + 16845556915121356121 p^{12} T^{16} + 94369533842064624 p^{15} T^{17} + 575954726100244 p^{18} T^{18} + 2673285031072 p^{21} T^{19} - 2568517965 p^{24} T^{20} - 62132888 p^{27} T^{21} - 109997 p^{30} T^{22} + 344 p^{33} T^{23} + p^{36} T^{24} \)
43 \( 1 + 512 T - 42346 T^{2} - 72172688 T^{3} - 13802808999 T^{4} - 25923143640 T^{5} + 163027793580538 T^{6} + 161446465996736192 T^{7} + \)\(12\!\cdots\!78\)\( T^{8} + \)\(37\!\cdots\!28\)\( T^{9} + \)\(51\!\cdots\!10\)\( T^{10} - \)\(26\!\cdots\!96\)\( T^{11} - \)\(98\!\cdots\!55\)\( T^{12} - \)\(26\!\cdots\!96\)\( p^{3} T^{13} + \)\(51\!\cdots\!10\)\( p^{6} T^{14} + \)\(37\!\cdots\!28\)\( p^{9} T^{15} + \)\(12\!\cdots\!78\)\( p^{12} T^{16} + 161446465996736192 p^{15} T^{17} + 163027793580538 p^{18} T^{18} - 25923143640 p^{21} T^{19} - 13802808999 p^{24} T^{20} - 72172688 p^{27} T^{21} - 42346 p^{30} T^{22} + 512 p^{33} T^{23} + p^{36} T^{24} \)
47 \( 1 + 238 T - 138371 T^{2} - 27559258 T^{3} + 21820300302 T^{4} + 3901648880510 T^{5} - 1082191078520777 T^{6} + 98919917267677062 T^{7} + 70759856644379366936 T^{8} - \)\(22\!\cdots\!18\)\( T^{9} + \)\(13\!\cdots\!65\)\( T^{10} + \)\(56\!\cdots\!22\)\( p T^{11} + \)\(96\!\cdots\!88\)\( T^{12} + \)\(56\!\cdots\!22\)\( p^{4} T^{13} + \)\(13\!\cdots\!65\)\( p^{6} T^{14} - \)\(22\!\cdots\!18\)\( p^{9} T^{15} + 70759856644379366936 p^{12} T^{16} + 98919917267677062 p^{15} T^{17} - 1082191078520777 p^{18} T^{18} + 3901648880510 p^{21} T^{19} + 21820300302 p^{24} T^{20} - 27559258 p^{27} T^{21} - 138371 p^{30} T^{22} + 238 p^{33} T^{23} + p^{36} T^{24} \)
53 \( ( 1 - 850 T + 343107 T^{2} - 31107110 T^{3} - 2956444769 T^{4} - 6953131975360 T^{5} + 6251628297772334 T^{6} - 6953131975360 p^{3} T^{7} - 2956444769 p^{6} T^{8} - 31107110 p^{9} T^{9} + 343107 p^{12} T^{10} - 850 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
59 \( 1 + 1840 T + 853879 T^{2} + 15427280 T^{3} + 373569815163 T^{4} + 428603646413600 T^{5} + 56251057684063156 T^{6} - 120477494103210240 p T^{7} + \)\(65\!\cdots\!49\)\( T^{8} + \)\(30\!\cdots\!80\)\( T^{9} - \)\(35\!\cdots\!51\)\( T^{10} + \)\(25\!\cdots\!20\)\( T^{11} + \)\(45\!\cdots\!86\)\( T^{12} + \)\(25\!\cdots\!20\)\( p^{3} T^{13} - \)\(35\!\cdots\!51\)\( p^{6} T^{14} + \)\(30\!\cdots\!80\)\( p^{9} T^{15} + \)\(65\!\cdots\!49\)\( p^{12} T^{16} - 120477494103210240 p^{16} T^{17} + 56251057684063156 p^{18} T^{18} + 428603646413600 p^{21} T^{19} + 373569815163 p^{24} T^{20} + 15427280 p^{27} T^{21} + 853879 p^{30} T^{22} + 1840 p^{33} T^{23} + p^{36} T^{24} \)
61 \( 1 - 364 T - 215302 T^{2} + 62471632 T^{3} - 64029627075 T^{4} + 36193606801512 T^{5} + 12986552131696030 T^{6} - 134223310243583980 p T^{7} + \)\(54\!\cdots\!14\)\( T^{8} - \)\(31\!\cdots\!12\)\( T^{9} - \)\(51\!\cdots\!42\)\( T^{10} + \)\(66\!\cdots\!52\)\( T^{11} - \)\(19\!\cdots\!47\)\( T^{12} + \)\(66\!\cdots\!52\)\( p^{3} T^{13} - \)\(51\!\cdots\!42\)\( p^{6} T^{14} - \)\(31\!\cdots\!12\)\( p^{9} T^{15} + \)\(54\!\cdots\!14\)\( p^{12} T^{16} - 134223310243583980 p^{16} T^{17} + 12986552131696030 p^{18} T^{18} + 36193606801512 p^{21} T^{19} - 64029627075 p^{24} T^{20} + 62471632 p^{27} T^{21} - 215302 p^{30} T^{22} - 364 p^{33} T^{23} + p^{36} T^{24} \)
67 \( 1 + 88 T - 483162 T^{2} + 357133536 T^{3} + 1427052415 p T^{4} - 114705539044208 T^{5} + 109045943557886754 T^{6} - 11312332891718112696 T^{7} - \)\(30\!\cdots\!74\)\( T^{8} + \)\(32\!\cdots\!08\)\( T^{9} - \)\(27\!\cdots\!66\)\( T^{10} - \)\(47\!\cdots\!84\)\( T^{11} + \)\(50\!\cdots\!81\)\( T^{12} - \)\(47\!\cdots\!84\)\( p^{3} T^{13} - \)\(27\!\cdots\!66\)\( p^{6} T^{14} + \)\(32\!\cdots\!08\)\( p^{9} T^{15} - \)\(30\!\cdots\!74\)\( p^{12} T^{16} - 11312332891718112696 p^{15} T^{17} + 109045943557886754 p^{18} T^{18} - 114705539044208 p^{21} T^{19} + 1427052415 p^{25} T^{20} + 357133536 p^{27} T^{21} - 483162 p^{30} T^{22} + 88 p^{33} T^{23} + p^{36} T^{24} \)
71 \( ( 1 - 1364 T + 1151472 T^{2} - 629760820 T^{3} + 269997702880 T^{4} - 18102353174084 T^{5} - 21873842560191886 T^{6} - 18102353174084 p^{3} T^{7} + 269997702880 p^{6} T^{8} - 629760820 p^{9} T^{9} + 1151472 p^{12} T^{10} - 1364 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
73 \( ( 1 - 836 T + 1203194 T^{2} - 845045868 T^{3} + 764405490259 T^{4} - 417329100597904 T^{5} + 337538834495072164 T^{6} - 417329100597904 p^{3} T^{7} + 764405490259 p^{6} T^{8} - 845045868 p^{9} T^{9} + 1203194 p^{12} T^{10} - 836 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
79 \( 1 - 680 T - 868038 T^{2} + 267439872 T^{3} + 521740485877 T^{4} + 88805897887120 T^{5} - 152622976676503218 T^{6} - \)\(15\!\cdots\!88\)\( T^{7} - \)\(12\!\cdots\!38\)\( T^{8} + \)\(53\!\cdots\!72\)\( T^{9} + \)\(70\!\cdots\!38\)\( T^{10} - \)\(59\!\cdots\!24\)\( T^{11} - \)\(51\!\cdots\!03\)\( T^{12} - \)\(59\!\cdots\!24\)\( p^{3} T^{13} + \)\(70\!\cdots\!38\)\( p^{6} T^{14} + \)\(53\!\cdots\!72\)\( p^{9} T^{15} - \)\(12\!\cdots\!38\)\( p^{12} T^{16} - \)\(15\!\cdots\!88\)\( p^{15} T^{17} - 152622976676503218 p^{18} T^{18} + 88805897887120 p^{21} T^{19} + 521740485877 p^{24} T^{20} + 267439872 p^{27} T^{21} - 868038 p^{30} T^{22} - 680 p^{33} T^{23} + p^{36} T^{24} \)
83 \( 1 + 2148 T + 528090 T^{2} - 1808793552 T^{3} - 612281780727 T^{4} + 1652483430830400 T^{5} + 851439447815137142 T^{6} - \)\(99\!\cdots\!08\)\( T^{7} - \)\(75\!\cdots\!50\)\( T^{8} + \)\(40\!\cdots\!36\)\( T^{9} + \)\(50\!\cdots\!74\)\( T^{10} - \)\(12\!\cdots\!64\)\( T^{11} - \)\(35\!\cdots\!99\)\( T^{12} - \)\(12\!\cdots\!64\)\( p^{3} T^{13} + \)\(50\!\cdots\!74\)\( p^{6} T^{14} + \)\(40\!\cdots\!36\)\( p^{9} T^{15} - \)\(75\!\cdots\!50\)\( p^{12} T^{16} - \)\(99\!\cdots\!08\)\( p^{15} T^{17} + 851439447815137142 p^{18} T^{18} + 1652483430830400 p^{21} T^{19} - 612281780727 p^{24} T^{20} - 1808793552 p^{27} T^{21} + 528090 p^{30} T^{22} + 2148 p^{33} T^{23} + p^{36} T^{24} \)
89 \( ( 1 - 3000 T + 6331152 T^{2} - 9569726760 T^{3} + 12126222621024 T^{4} - 12808596417843000 T^{5} + 11672528635101585634 T^{6} - 12808596417843000 p^{3} T^{7} + 12126222621024 p^{6} T^{8} - 9569726760 p^{9} T^{9} + 6331152 p^{12} T^{10} - 3000 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
97 \( 1 - 612 T - 1933782 T^{2} + 1082142976 T^{3} + 453155126529 T^{4} + 311147317433664 T^{5} - 662021276643806282 T^{6} - \)\(87\!\cdots\!56\)\( T^{7} + \)\(35\!\cdots\!18\)\( T^{8} - \)\(11\!\cdots\!44\)\( T^{9} - \)\(23\!\cdots\!06\)\( T^{10} + \)\(12\!\cdots\!80\)\( T^{11} + \)\(50\!\cdots\!09\)\( T^{12} + \)\(12\!\cdots\!80\)\( p^{3} T^{13} - \)\(23\!\cdots\!06\)\( p^{6} T^{14} - \)\(11\!\cdots\!44\)\( p^{9} T^{15} + \)\(35\!\cdots\!18\)\( p^{12} T^{16} - \)\(87\!\cdots\!56\)\( p^{15} T^{17} - 662021276643806282 p^{18} T^{18} + 311147317433664 p^{21} T^{19} + 453155126529 p^{24} T^{20} + 1082142976 p^{27} T^{21} - 1933782 p^{30} T^{22} - 612 p^{33} T^{23} + p^{36} T^{24} \)
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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

−3.22726825467382469934992497052, −3.22396592514390922306725674005, −3.14723168362126079989478634634, −3.01053959519038441884542041925, −2.94255531208329593985729413811, −2.59624352281687024516706137479, −2.53148285091781437053483612048, −2.48934643509059807346059533151, −2.43378700435334734170608993485, −2.07450329355344618834840184595, −1.99627079106769046998292235380, −1.96538845222514004466506968513, −1.87841066702388538803981226599, −1.82957331246775267430878284376, −1.74049013693790037521381984840, −1.39011420775138007015408401250, −1.12704334570821606898504555551, −1.08051972773218010881653839914, −0.867579788762575429703371294669, −0.72109176007278871045405310221, −0.64512984861100292514085925163, −0.55379698533793517296256809791, −0.34501334448093173358856501434, −0.22872869658591736297526059245, −0.07103975294539573573797852231, 0.07103975294539573573797852231, 0.22872869658591736297526059245, 0.34501334448093173358856501434, 0.55379698533793517296256809791, 0.64512984861100292514085925163, 0.72109176007278871045405310221, 0.867579788762575429703371294669, 1.08051972773218010881653839914, 1.12704334570821606898504555551, 1.39011420775138007015408401250, 1.74049013693790037521381984840, 1.82957331246775267430878284376, 1.87841066702388538803981226599, 1.96538845222514004466506968513, 1.99627079106769046998292235380, 2.07450329355344618834840184595, 2.43378700435334734170608993485, 2.48934643509059807346059533151, 2.53148285091781437053483612048, 2.59624352281687024516706137479, 2.94255531208329593985729413811, 3.01053959519038441884542041925, 3.14723168362126079989478634634, 3.22396592514390922306725674005, 3.22726825467382469934992497052

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

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