Properties

Label 24-666e12-1.1-c1e12-0-2
Degree $24$
Conductor $7.615\times 10^{33}$
Sign $1$
Analytic cond. $5.11700\times 10^{8}$
Root an. cond. $2.30608$
Motivic weight $1$
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  − 12·7-s + 6·11-s + 6·13-s − 18·19-s − 9·25-s − 18·29-s + 30·37-s − 24·41-s − 6·47-s + 78·49-s + 12·53-s − 36·61-s + 64-s − 30·67-s − 12·71-s − 72·77-s + 6·79-s + 48·83-s + 18·89-s − 72·91-s + 36·97-s + 24·101-s + 36·103-s − 90·107-s + 6·109-s + 39·121-s + 18·125-s + ⋯
L(s)  = 1  − 4.53·7-s + 1.80·11-s + 1.66·13-s − 4.12·19-s − 9/5·25-s − 3.34·29-s + 4.93·37-s − 3.74·41-s − 0.875·47-s + 78/7·49-s + 1.64·53-s − 4.60·61-s + 1/8·64-s − 3.66·67-s − 1.42·71-s − 8.20·77-s + 0.675·79-s + 5.26·83-s + 1.90·89-s − 7.54·91-s + 3.65·97-s + 2.38·101-s + 3.54·103-s − 8.70·107-s + 0.574·109-s + 3.54·121-s + 1.60·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{12} \cdot 3^{24} \cdot 37^{12}\)
Sign: $1$
Analytic conductor: \(5.11700\times 10^{8}\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{12} \cdot 3^{24} \cdot 37^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9549788063\)
\(L(\frac12)\) \(\approx\) \(0.9549788063\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{6} + T^{12} \)
3 \( 1 \)
37 \( 1 - 30 T + 417 T^{2} - 3838 T^{3} + 27450 T^{4} - 164430 T^{5} + 953241 T^{6} - 164430 p T^{7} + 27450 p^{2} T^{8} - 3838 p^{3} T^{9} + 417 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
good5 \( 1 + 9 T^{2} - 18 T^{3} + 36 T^{4} - 108 T^{5} + 286 T^{6} - 162 T^{7} + 351 T^{8} - 252 p T^{9} - 5589 T^{10} + 1242 p T^{11} - 22119 T^{12} + 1242 p^{2} T^{13} - 5589 p^{2} T^{14} - 252 p^{4} T^{15} + 351 p^{4} T^{16} - 162 p^{5} T^{17} + 286 p^{6} T^{18} - 108 p^{7} T^{19} + 36 p^{8} T^{20} - 18 p^{9} T^{21} + 9 p^{10} T^{22} + p^{12} T^{24} \)
7 \( 1 + 12 T + 66 T^{2} + 218 T^{3} + 66 p T^{4} + 12 p^{2} T^{5} + 289 T^{6} - 648 T^{7} - 4131 T^{8} - 22354 T^{9} - 327 p^{3} T^{10} - 455736 T^{11} - 1395491 T^{12} - 455736 p T^{13} - 327 p^{5} T^{14} - 22354 p^{3} T^{15} - 4131 p^{4} T^{16} - 648 p^{5} T^{17} + 289 p^{6} T^{18} + 12 p^{9} T^{19} + 66 p^{9} T^{20} + 218 p^{9} T^{21} + 66 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 - 6 T - 3 T^{2} + 126 T^{3} - 384 T^{4} - 78 T^{5} + 3947 T^{6} - 14724 T^{7} + 20151 T^{8} + 35478 T^{9} - 185580 T^{10} + 507132 T^{11} - 1637679 T^{12} + 507132 p T^{13} - 185580 p^{2} T^{14} + 35478 p^{3} T^{15} + 20151 p^{4} T^{16} - 14724 p^{5} T^{17} + 3947 p^{6} T^{18} - 78 p^{7} T^{19} - 384 p^{8} T^{20} + 126 p^{9} T^{21} - 3 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 - 6 T + 30 T^{2} + 60 T^{3} - 912 T^{4} + 5154 T^{5} - 8551 T^{6} - 47682 T^{7} + 30951 p T^{8} - 1393602 T^{9} + 47223 p T^{10} + 15564222 T^{11} - 89154855 T^{12} + 15564222 p T^{13} + 47223 p^{3} T^{14} - 1393602 p^{3} T^{15} + 30951 p^{5} T^{16} - 47682 p^{5} T^{17} - 8551 p^{6} T^{18} + 5154 p^{7} T^{19} - 912 p^{8} T^{20} + 60 p^{9} T^{21} + 30 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 + 36 T^{2} + 72 T^{3} + 576 T^{4} + 432 T^{5} + 7810 T^{6} - 17496 T^{7} - 83268 T^{8} - 7632 p T^{9} - 3484188 T^{10} - 6754320 T^{11} - 47006925 T^{12} - 6754320 p T^{13} - 3484188 p^{2} T^{14} - 7632 p^{4} T^{15} - 83268 p^{4} T^{16} - 17496 p^{5} T^{17} + 7810 p^{6} T^{18} + 432 p^{7} T^{19} + 576 p^{8} T^{20} + 72 p^{9} T^{21} + 36 p^{10} T^{22} + p^{12} T^{24} \)
19 \( 1 + 18 T + 135 T^{2} + 36 p T^{3} + 3987 T^{4} + 22842 T^{5} + 86699 T^{6} + 237294 T^{7} + 682254 T^{8} + 221364 T^{9} - 17657496 T^{10} - 114009210 T^{11} - 491591103 T^{12} - 114009210 p T^{13} - 17657496 p^{2} T^{14} + 221364 p^{3} T^{15} + 682254 p^{4} T^{16} + 237294 p^{5} T^{17} + 86699 p^{6} T^{18} + 22842 p^{7} T^{19} + 3987 p^{8} T^{20} + 36 p^{10} T^{21} + 135 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 + 75 T^{2} + 2820 T^{4} + 486 T^{5} + 2843 p T^{6} + 88938 T^{7} + 1000341 T^{8} + 5307120 T^{9} + 9514314 T^{10} + 187937658 T^{11} + 84111261 T^{12} + 187937658 p T^{13} + 9514314 p^{2} T^{14} + 5307120 p^{3} T^{15} + 1000341 p^{4} T^{16} + 88938 p^{5} T^{17} + 2843 p^{7} T^{18} + 486 p^{7} T^{19} + 2820 p^{8} T^{20} + 75 p^{10} T^{22} + p^{12} T^{24} \)
29 \( 1 + 18 T + 300 T^{2} + 3456 T^{3} + 37074 T^{4} + 328356 T^{5} + 2747362 T^{6} + 20241414 T^{7} + 142598466 T^{8} + 913009860 T^{9} + 5656241286 T^{10} + 32404114530 T^{11} + 180882283299 T^{12} + 32404114530 p T^{13} + 5656241286 p^{2} T^{14} + 913009860 p^{3} T^{15} + 142598466 p^{4} T^{16} + 20241414 p^{5} T^{17} + 2747362 p^{6} T^{18} + 328356 p^{7} T^{19} + 37074 p^{8} T^{20} + 3456 p^{9} T^{21} + 300 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 - 162 T^{2} + 15255 T^{4} - 1015396 T^{6} + 52463889 T^{8} - 2178592578 T^{10} + 74378126991 T^{12} - 2178592578 p^{2} T^{14} + 52463889 p^{4} T^{16} - 1015396 p^{6} T^{18} + 15255 p^{8} T^{20} - 162 p^{10} T^{22} + p^{12} T^{24} \)
41 \( 1 + 24 T + 216 T^{2} + 828 T^{3} + 720 T^{4} + 13956 T^{5} + 391898 T^{6} + 3438108 T^{7} + 10896192 T^{8} - 13992048 T^{9} + 77900472 T^{10} + 5170021092 T^{11} + 50104727283 T^{12} + 5170021092 p T^{13} + 77900472 p^{2} T^{14} - 13992048 p^{3} T^{15} + 10896192 p^{4} T^{16} + 3438108 p^{5} T^{17} + 391898 p^{6} T^{18} + 13956 p^{7} T^{19} + 720 p^{8} T^{20} + 828 p^{9} T^{21} + 216 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 - 360 T^{2} + 63864 T^{4} - 7338454 T^{6} + 606798000 T^{8} - 38000588400 T^{10} + 1847557538571 T^{12} - 38000588400 p^{2} T^{14} + 606798000 p^{4} T^{16} - 7338454 p^{6} T^{18} + 63864 p^{8} T^{20} - 360 p^{10} T^{22} + p^{12} T^{24} \)
47 \( 1 + 6 T - 129 T^{2} - 630 T^{3} + 7275 T^{4} + 13200 T^{5} - 10496 p T^{6} + 177264 T^{7} + 38861325 T^{8} + 36485370 T^{9} - 1800081855 T^{10} - 37175094 p T^{11} + 66999936462 T^{12} - 37175094 p^{2} T^{13} - 1800081855 p^{2} T^{14} + 36485370 p^{3} T^{15} + 38861325 p^{4} T^{16} + 177264 p^{5} T^{17} - 10496 p^{7} T^{18} + 13200 p^{7} T^{19} + 7275 p^{8} T^{20} - 630 p^{9} T^{21} - 129 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 - 12 T + 162 T^{2} - 2070 T^{3} + 17406 T^{4} - 184260 T^{5} + 1542269 T^{6} - 11580300 T^{7} + 102370437 T^{8} - 674033346 T^{9} + 5148381051 T^{10} - 39964816152 T^{11} + 251484462021 T^{12} - 39964816152 p T^{13} + 5148381051 p^{2} T^{14} - 674033346 p^{3} T^{15} + 102370437 p^{4} T^{16} - 11580300 p^{5} T^{17} + 1542269 p^{6} T^{18} - 184260 p^{7} T^{19} + 17406 p^{8} T^{20} - 2070 p^{9} T^{21} + 162 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 - 342 T^{3} + 1674 T^{4} + 49014 T^{5} - 53525 T^{6} - 79740 T^{7} - 16674903 T^{8} + 113688882 T^{9} + 1369196325 T^{10} - 3553369002 T^{11} - 10101568623 T^{12} - 3553369002 p T^{13} + 1369196325 p^{2} T^{14} + 113688882 p^{3} T^{15} - 16674903 p^{4} T^{16} - 79740 p^{5} T^{17} - 53525 p^{6} T^{18} + 49014 p^{7} T^{19} + 1674 p^{8} T^{20} - 342 p^{9} T^{21} + p^{12} T^{24} \)
61 \( 1 + 36 T + 756 T^{2} + 12564 T^{3} + 176328 T^{4} + 2189844 T^{5} + 24855698 T^{6} + 261642168 T^{7} + 2584948968 T^{8} + 24136234056 T^{9} + 213689013264 T^{10} + 1797416529192 T^{11} + 14385250976619 T^{12} + 1797416529192 p T^{13} + 213689013264 p^{2} T^{14} + 24136234056 p^{3} T^{15} + 2584948968 p^{4} T^{16} + 261642168 p^{5} T^{17} + 24855698 p^{6} T^{18} + 2189844 p^{7} T^{19} + 176328 p^{8} T^{20} + 12564 p^{9} T^{21} + 756 p^{10} T^{22} + 36 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 30 T + 669 T^{2} + 10784 T^{3} + 145236 T^{4} + 1633170 T^{5} + 15989248 T^{6} + 136598616 T^{7} + 1029520098 T^{8} + 6927400208 T^{9} + 42351918369 T^{10} + 258461868804 T^{11} + 1824238562935 T^{12} + 258461868804 p T^{13} + 42351918369 p^{2} T^{14} + 6927400208 p^{3} T^{15} + 1029520098 p^{4} T^{16} + 136598616 p^{5} T^{17} + 15989248 p^{6} T^{18} + 1633170 p^{7} T^{19} + 145236 p^{8} T^{20} + 10784 p^{9} T^{21} + 669 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 + 12 T + 216 T^{2} + 2232 T^{3} + 360 p T^{4} + 171552 T^{5} + 1359938 T^{6} + 7875324 T^{7} + 67831200 T^{8} + 510242112 T^{9} + 8171477388 T^{10} + 73279337940 T^{11} + 777801762579 T^{12} + 73279337940 p T^{13} + 8171477388 p^{2} T^{14} + 510242112 p^{3} T^{15} + 67831200 p^{4} T^{16} + 7875324 p^{5} T^{17} + 1359938 p^{6} T^{18} + 171552 p^{7} T^{19} + 360 p^{9} T^{20} + 2232 p^{9} T^{21} + 216 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \)
73 \( ( 1 + 72 T^{2} + 322 T^{3} + 1368 T^{4} - 44892 T^{5} + 304995 T^{6} - 44892 p T^{7} + 1368 p^{2} T^{8} + 322 p^{3} T^{9} + 72 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( 1 - 6 T + 12 T^{2} + 1248 T^{3} - 10650 T^{4} - 22728 T^{5} + 1578554 T^{6} - 11024046 T^{7} - 6224922 T^{8} + 1317987588 T^{9} - 6433288974 T^{10} - 46382041998 T^{11} + 1018948827267 T^{12} - 46382041998 p T^{13} - 6433288974 p^{2} T^{14} + 1317987588 p^{3} T^{15} - 6224922 p^{4} T^{16} - 11024046 p^{5} T^{17} + 1578554 p^{6} T^{18} - 22728 p^{7} T^{19} - 10650 p^{8} T^{20} + 1248 p^{9} T^{21} + 12 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 - 48 T + 1008 T^{2} - 11466 T^{3} + 60894 T^{4} + 91014 T^{5} - 1749697 T^{6} - 34462044 T^{7} + 8868375 p T^{8} - 6180808302 T^{9} + 30987329049 T^{10} - 170074677726 T^{11} + 1524826340769 T^{12} - 170074677726 p T^{13} + 30987329049 p^{2} T^{14} - 6180808302 p^{3} T^{15} + 8868375 p^{5} T^{16} - 34462044 p^{5} T^{17} - 1749697 p^{6} T^{18} + 91014 p^{7} T^{19} + 60894 p^{8} T^{20} - 11466 p^{9} T^{21} + 1008 p^{10} T^{22} - 48 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 - 18 T + 9 T^{2} + 1296 T^{3} + 16236 T^{4} - 284814 T^{5} - 1490888 T^{6} + 24667128 T^{7} + 210783978 T^{8} - 1425964500 T^{9} - 29432650887 T^{10} + 99080020404 T^{11} + 2142973122519 T^{12} + 99080020404 p T^{13} - 29432650887 p^{2} T^{14} - 1425964500 p^{3} T^{15} + 210783978 p^{4} T^{16} + 24667128 p^{5} T^{17} - 1490888 p^{6} T^{18} - 284814 p^{7} T^{19} + 16236 p^{8} T^{20} + 1296 p^{9} T^{21} + 9 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 - 36 T + 918 T^{2} - 17496 T^{3} + 280341 T^{4} - 3647664 T^{5} + 39168002 T^{6} - 317393748 T^{7} + 1369191978 T^{8} + 12571294932 T^{9} - 398485202130 T^{10} + 6007641045408 T^{11} - 66848579588931 T^{12} + 6007641045408 p T^{13} - 398485202130 p^{2} T^{14} + 12571294932 p^{3} T^{15} + 1369191978 p^{4} T^{16} - 317393748 p^{5} T^{17} + 39168002 p^{6} T^{18} - 3647664 p^{7} T^{19} + 280341 p^{8} T^{20} - 17496 p^{9} T^{21} + 918 p^{10} T^{22} - 36 p^{11} T^{23} + p^{12} T^{24} \)
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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

−3.43684369406898987111630978775, −3.35748695309881018635750581864, −3.32825285484429288238902194958, −3.05751545129879217897027002828, −3.04664155857620972791245791439, −2.96423298236873727139509668058, −2.91814945480366620069647992350, −2.77502915926724515891825867208, −2.69443190832485192777184981262, −2.58367649238597973120725074239, −2.31780008317526002909543174989, −2.16279481758778609817008404746, −2.11115873107271349391616793604, −2.00626002608542527129342900166, −1.85613112869709213880136028256, −1.84779077836505118556444181594, −1.75832955285474708364841631685, −1.63815673782269844590108289842, −1.34881592513410633704648068195, −1.15302491183796279839562357295, −0.972805713065164850700429362278, −0.76215683280277522638336837626, −0.42104885896602155377247312217, −0.31584257668796551679184125376, −0.25389976728869245259753105109, 0.25389976728869245259753105109, 0.31584257668796551679184125376, 0.42104885896602155377247312217, 0.76215683280277522638336837626, 0.972805713065164850700429362278, 1.15302491183796279839562357295, 1.34881592513410633704648068195, 1.63815673782269844590108289842, 1.75832955285474708364841631685, 1.84779077836505118556444181594, 1.85613112869709213880136028256, 2.00626002608542527129342900166, 2.11115873107271349391616793604, 2.16279481758778609817008404746, 2.31780008317526002909543174989, 2.58367649238597973120725074239, 2.69443190832485192777184981262, 2.77502915926724515891825867208, 2.91814945480366620069647992350, 2.96423298236873727139509668058, 3.04664155857620972791245791439, 3.05751545129879217897027002828, 3.32825285484429288238902194958, 3.35748695309881018635750581864, 3.43684369406898987111630978775

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

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