Properties

Degree $24$
Conductor $6.913\times 10^{46}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $12$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s − 3·5-s + 4·7-s + 78·9-s − 5·11-s − 6·13-s + 36·15-s − 7·17-s − 3·19-s − 48·21-s + 12·23-s − 20·25-s − 364·27-s − 12·29-s + 2·31-s + 60·33-s − 12·35-s − 20·37-s + 72·39-s − 3·41-s + 5·43-s − 234·45-s − 35·49-s + 84·51-s − 3·53-s + 15·55-s + 36·57-s + ⋯
L(s)  = 1  − 6.92·3-s − 1.34·5-s + 1.51·7-s + 26·9-s − 1.50·11-s − 1.66·13-s + 9.29·15-s − 1.69·17-s − 0.688·19-s − 10.4·21-s + 2.50·23-s − 4·25-s − 70.0·27-s − 2.22·29-s + 0.359·31-s + 10.4·33-s − 2.02·35-s − 3.28·37-s + 11.5·39-s − 0.468·41-s + 0.762·43-s − 34.8·45-s − 5·49-s + 11.7·51-s − 0.412·53-s + 2.02·55-s + 4.76·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 23^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 23^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 23^{12} \cdot 29^{12}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{8004} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(12\)
Selberg data: \((24,\ 2^{24} \cdot 3^{12} \cdot 23^{12} \cdot 29^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T )^{12} \)
23 \( ( 1 - T )^{12} \)
29 \( ( 1 + T )^{12} \)
good5 \( 1 + 3 T + 29 T^{2} + 17 p T^{3} + 447 T^{4} + 1222 T^{5} + 4791 T^{6} + 12124 T^{7} + 39159 T^{8} + 92181 T^{9} + 257964 T^{10} + 561329 T^{11} + 1410706 T^{12} + 561329 p T^{13} + 257964 p^{2} T^{14} + 92181 p^{3} T^{15} + 39159 p^{4} T^{16} + 12124 p^{5} T^{17} + 4791 p^{6} T^{18} + 1222 p^{7} T^{19} + 447 p^{8} T^{20} + 17 p^{10} T^{21} + 29 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
7 \( 1 - 4 T + 51 T^{2} - 181 T^{3} + 1306 T^{4} - 4099 T^{5} + 3130 p T^{6} - 61538 T^{7} + 269911 T^{8} - 683906 T^{9} + 2595787 T^{10} - 5956898 T^{11} + 2875364 p T^{12} - 5956898 p T^{13} + 2595787 p^{2} T^{14} - 683906 p^{3} T^{15} + 269911 p^{4} T^{16} - 61538 p^{5} T^{17} + 3130 p^{7} T^{18} - 4099 p^{7} T^{19} + 1306 p^{8} T^{20} - 181 p^{9} T^{21} + 51 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 + 5 T + 84 T^{2} + 394 T^{3} + 3529 T^{4} + 1383 p T^{5} + 98145 T^{6} + 34818 p T^{7} + 2001529 T^{8} + 7024517 T^{9} + 2853897 p T^{10} + 98769211 T^{11} + 387711866 T^{12} + 98769211 p T^{13} + 2853897 p^{3} T^{14} + 7024517 p^{3} T^{15} + 2001529 p^{4} T^{16} + 34818 p^{6} T^{17} + 98145 p^{6} T^{18} + 1383 p^{8} T^{19} + 3529 p^{8} T^{20} + 394 p^{9} T^{21} + 84 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 + 6 T + 103 T^{2} + 545 T^{3} + 4809 T^{4} + 22421 T^{5} + 135951 T^{6} + 557101 T^{7} + 2662315 T^{8} + 9670399 T^{9} + 40531145 T^{10} + 135620044 T^{11} + 543448756 T^{12} + 135620044 p T^{13} + 40531145 p^{2} T^{14} + 9670399 p^{3} T^{15} + 2662315 p^{4} T^{16} + 557101 p^{5} T^{17} + 135951 p^{6} T^{18} + 22421 p^{7} T^{19} + 4809 p^{8} T^{20} + 545 p^{9} T^{21} + 103 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 + 7 T + 135 T^{2} + 779 T^{3} + 8739 T^{4} + 43410 T^{5} + 367973 T^{6} + 1616106 T^{7} + 11359339 T^{8} + 44687859 T^{9} + 271762001 T^{10} + 960357097 T^{11} + 5165105850 T^{12} + 960357097 p T^{13} + 271762001 p^{2} T^{14} + 44687859 p^{3} T^{15} + 11359339 p^{4} T^{16} + 1616106 p^{5} T^{17} + 367973 p^{6} T^{18} + 43410 p^{7} T^{19} + 8739 p^{8} T^{20} + 779 p^{9} T^{21} + 135 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 + 3 T + 158 T^{2} + 469 T^{3} + 12128 T^{4} + 35189 T^{5} + 600218 T^{6} + 1677275 T^{7} + 59344 p^{2} T^{8} + 56533655 T^{9} + 583298614 T^{10} + 1416071249 T^{11} + 12458946078 T^{12} + 1416071249 p T^{13} + 583298614 p^{2} T^{14} + 56533655 p^{3} T^{15} + 59344 p^{6} T^{16} + 1677275 p^{5} T^{17} + 600218 p^{6} T^{18} + 35189 p^{7} T^{19} + 12128 p^{8} T^{20} + 469 p^{9} T^{21} + 158 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 - 2 T + 170 T^{2} - 137 T^{3} + 15770 T^{4} - 5574 T^{5} + 1044154 T^{6} - 101845 T^{7} + 53100647 T^{8} + 2363693 T^{9} + 2184198652 T^{10} + 201912019 T^{11} + 74253191996 T^{12} + 201912019 p T^{13} + 2184198652 p^{2} T^{14} + 2363693 p^{3} T^{15} + 53100647 p^{4} T^{16} - 101845 p^{5} T^{17} + 1044154 p^{6} T^{18} - 5574 p^{7} T^{19} + 15770 p^{8} T^{20} - 137 p^{9} T^{21} + 170 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 + 20 T + 446 T^{2} + 5603 T^{3} + 73349 T^{4} + 690156 T^{5} + 6833206 T^{6} + 53316724 T^{7} + 447546425 T^{8} + 3088469653 T^{9} + 23080086808 T^{10} + 143320744868 T^{11} + 956581799354 T^{12} + 143320744868 p T^{13} + 23080086808 p^{2} T^{14} + 3088469653 p^{3} T^{15} + 447546425 p^{4} T^{16} + 53316724 p^{5} T^{17} + 6833206 p^{6} T^{18} + 690156 p^{7} T^{19} + 73349 p^{8} T^{20} + 5603 p^{9} T^{21} + 446 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 + 3 T + 5 p T^{2} + 307 T^{3} + 18175 T^{4} + 340 T^{5} + 936847 T^{6} - 738690 T^{7} + 37490859 T^{8} + 33321383 T^{9} + 1662380140 T^{10} + 6442561697 T^{11} + 74723330234 T^{12} + 6442561697 p T^{13} + 1662380140 p^{2} T^{14} + 33321383 p^{3} T^{15} + 37490859 p^{4} T^{16} - 738690 p^{5} T^{17} + 936847 p^{6} T^{18} + 340 p^{7} T^{19} + 18175 p^{8} T^{20} + 307 p^{9} T^{21} + 5 p^{11} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 - 5 T + 227 T^{2} - 391 T^{3} + 21416 T^{4} + 29505 T^{5} + 1334662 T^{6} + 5130395 T^{7} + 78487526 T^{8} + 307723151 T^{9} + 4639561771 T^{10} + 12327703765 T^{11} + 228789635290 T^{12} + 12327703765 p T^{13} + 4639561771 p^{2} T^{14} + 307723151 p^{3} T^{15} + 78487526 p^{4} T^{16} + 5130395 p^{5} T^{17} + 1334662 p^{6} T^{18} + 29505 p^{7} T^{19} + 21416 p^{8} T^{20} - 391 p^{9} T^{21} + 227 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 + 302 T^{2} + 722 T^{3} + 45483 T^{4} + 191777 T^{5} + 4737251 T^{6} + 24986927 T^{7} + 382331725 T^{8} + 2126276463 T^{9} + 24783876435 T^{10} + 131567343855 T^{11} + 1297056605334 T^{12} + 131567343855 p T^{13} + 24783876435 p^{2} T^{14} + 2126276463 p^{3} T^{15} + 382331725 p^{4} T^{16} + 24986927 p^{5} T^{17} + 4737251 p^{6} T^{18} + 191777 p^{7} T^{19} + 45483 p^{8} T^{20} + 722 p^{9} T^{21} + 302 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 + 3 T + 369 T^{2} + 869 T^{3} + 68815 T^{4} + 142618 T^{5} + 8630531 T^{6} + 16558800 T^{7} + 805876379 T^{8} + 1442902733 T^{9} + 58986949764 T^{10} + 97648128425 T^{11} + 3473322956874 T^{12} + 97648128425 p T^{13} + 58986949764 p^{2} T^{14} + 1442902733 p^{3} T^{15} + 805876379 p^{4} T^{16} + 16558800 p^{5} T^{17} + 8630531 p^{6} T^{18} + 142618 p^{7} T^{19} + 68815 p^{8} T^{20} + 869 p^{9} T^{21} + 369 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 + 20 T + 494 T^{2} + 5912 T^{3} + 87243 T^{4} + 765416 T^{5} + 9306170 T^{6} + 70130916 T^{7} + 812437859 T^{8} + 5683683792 T^{9} + 61995753224 T^{10} + 398433959328 T^{11} + 3990056201442 T^{12} + 398433959328 p T^{13} + 61995753224 p^{2} T^{14} + 5683683792 p^{3} T^{15} + 812437859 p^{4} T^{16} + 70130916 p^{5} T^{17} + 9306170 p^{6} T^{18} + 765416 p^{7} T^{19} + 87243 p^{8} T^{20} + 5912 p^{9} T^{21} + 494 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 + 17 T + 510 T^{2} + 6804 T^{3} + 120491 T^{4} + 1365399 T^{5} + 18372002 T^{6} + 183570674 T^{7} + 2050230403 T^{8} + 18314912721 T^{9} + 176756642032 T^{10} + 1415754289261 T^{11} + 197836496010 p T^{12} + 1415754289261 p T^{13} + 176756642032 p^{2} T^{14} + 18314912721 p^{3} T^{15} + 2050230403 p^{4} T^{16} + 183570674 p^{5} T^{17} + 18372002 p^{6} T^{18} + 1365399 p^{7} T^{19} + 120491 p^{8} T^{20} + 6804 p^{9} T^{21} + 510 p^{10} T^{22} + 17 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 9 T + 563 T^{2} + 4525 T^{3} + 147688 T^{4} + 1072099 T^{5} + 24581492 T^{6} + 162558481 T^{7} + 2972408127 T^{8} + 17933807816 T^{9} + 279171589761 T^{10} + 1526212276726 T^{11} + 20896286012256 T^{12} + 1526212276726 p T^{13} + 279171589761 p^{2} T^{14} + 17933807816 p^{3} T^{15} + 2972408127 p^{4} T^{16} + 162558481 p^{5} T^{17} + 24581492 p^{6} T^{18} + 1072099 p^{7} T^{19} + 147688 p^{8} T^{20} + 4525 p^{9} T^{21} + 563 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 - 7 T + 476 T^{2} - 1492 T^{3} + 95284 T^{4} + 66242 T^{5} + 11768912 T^{6} + 49115224 T^{7} + 1226805976 T^{8} + 6523649352 T^{9} + 122762751080 T^{10} + 519837843019 T^{11} + 10106802962142 T^{12} + 519837843019 p T^{13} + 122762751080 p^{2} T^{14} + 6523649352 p^{3} T^{15} + 1226805976 p^{4} T^{16} + 49115224 p^{5} T^{17} + 11768912 p^{6} T^{18} + 66242 p^{7} T^{19} + 95284 p^{8} T^{20} - 1492 p^{9} T^{21} + 476 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 + 9 T + 709 T^{2} + 4662 T^{3} + 220376 T^{4} + 965723 T^{5} + 40155885 T^{6} + 92349544 T^{7} + 4908938519 T^{8} + 2009273495 T^{9} + 450446125726 T^{10} - 377142877857 T^{11} + 34808677030176 T^{12} - 377142877857 p T^{13} + 450446125726 p^{2} T^{14} + 2009273495 p^{3} T^{15} + 4908938519 p^{4} T^{16} + 92349544 p^{5} T^{17} + 40155885 p^{6} T^{18} + 965723 p^{7} T^{19} + 220376 p^{8} T^{20} + 4662 p^{9} T^{21} + 709 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 - 14 T + 410 T^{2} - 4613 T^{3} + 78351 T^{4} - 792764 T^{5} + 11062290 T^{6} - 109467202 T^{7} + 1344986159 T^{8} - 12514380775 T^{9} + 133618743464 T^{10} - 1142424924076 T^{11} + 11165347207894 T^{12} - 1142424924076 p T^{13} + 133618743464 p^{2} T^{14} - 12514380775 p^{3} T^{15} + 1344986159 p^{4} T^{16} - 109467202 p^{5} T^{17} + 11062290 p^{6} T^{18} - 792764 p^{7} T^{19} + 78351 p^{8} T^{20} - 4613 p^{9} T^{21} + 410 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 - 5 T + 565 T^{2} - 923 T^{3} + 151223 T^{4} + 97770 T^{5} + 28057219 T^{6} + 49140288 T^{7} + 4066956867 T^{8} + 8728187813 T^{9} + 465226469736 T^{10} + 1049244370717 T^{11} + 42722631463610 T^{12} + 1049244370717 p T^{13} + 465226469736 p^{2} T^{14} + 8728187813 p^{3} T^{15} + 4066956867 p^{4} T^{16} + 49140288 p^{5} T^{17} + 28057219 p^{6} T^{18} + 97770 p^{7} T^{19} + 151223 p^{8} T^{20} - 923 p^{9} T^{21} + 565 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 + 22 T + 911 T^{2} + 16895 T^{3} + 390631 T^{4} + 6201311 T^{5} + 104449155 T^{6} + 1434798065 T^{7} + 19412628579 T^{8} + 232283103135 T^{9} + 2643023184249 T^{10} + 27607421552446 T^{11} + 270352082987194 T^{12} + 27607421552446 p T^{13} + 2643023184249 p^{2} T^{14} + 232283103135 p^{3} T^{15} + 19412628579 p^{4} T^{16} + 1434798065 p^{5} T^{17} + 104449155 p^{6} T^{18} + 6201311 p^{7} T^{19} + 390631 p^{8} T^{20} + 16895 p^{9} T^{21} + 911 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 - 17 T + 774 T^{2} - 9880 T^{3} + 253537 T^{4} - 2584489 T^{5} + 503712 p T^{6} - 416211196 T^{7} + 67168095 p T^{8} - 48877804967 T^{9} + 691371657370 T^{10} - 4897327112811 T^{11} + 67453204211902 T^{12} - 4897327112811 p T^{13} + 691371657370 p^{2} T^{14} - 48877804967 p^{3} T^{15} + 67168095 p^{5} T^{16} - 416211196 p^{5} T^{17} + 503712 p^{7} T^{18} - 2584489 p^{7} T^{19} + 253537 p^{8} T^{20} - 9880 p^{9} T^{21} + 774 p^{10} T^{22} - 17 p^{11} T^{23} + p^{12} T^{24} \)
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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

−2.78646473486968306180134515911, −2.37223667776865014923104019748, −2.35695805378065866472814312627, −2.31549568410892928583216848976, −2.28391241108589756952100231907, −2.26523177655415254199657651043, −2.24369331993859501726431566492, −2.10309108356817580903630521657, −2.09500500187687788701893460618, −2.00856249862205444391266130418, −1.95649843065576687548074786143, −1.94039604880786427856304263683, −1.90451609430070012392623032903, −1.61551138226079428243645151775, −1.36880932160754387390361150958, −1.35485314070441286542663322758, −1.28718439210837633325223689646, −1.26991776593545882670296912952, −1.24212863037046833228621667519, −1.19367625317729728681209256683, −1.18222793431395782091392522428, −1.04605010374714189542577855076, −1.04022949183673160588447109748, −0.895910024536567704333192372342, −0.807268740114146943614618514108, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.807268740114146943614618514108, 0.895910024536567704333192372342, 1.04022949183673160588447109748, 1.04605010374714189542577855076, 1.18222793431395782091392522428, 1.19367625317729728681209256683, 1.24212863037046833228621667519, 1.26991776593545882670296912952, 1.28718439210837633325223689646, 1.35485314070441286542663322758, 1.36880932160754387390361150958, 1.61551138226079428243645151775, 1.90451609430070012392623032903, 1.94039604880786427856304263683, 1.95649843065576687548074786143, 2.00856249862205444391266130418, 2.09500500187687788701893460618, 2.10309108356817580903630521657, 2.24369331993859501726431566492, 2.26523177655415254199657651043, 2.28391241108589756952100231907, 2.31549568410892928583216848976, 2.35695805378065866472814312627, 2.37223667776865014923104019748, 2.78646473486968306180134515911

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

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