Properties

Label 24-42e12-1.1-c8e12-0-1
Degree $24$
Conductor $3.013\times 10^{19}$
Sign $1$
Analytic cond. $6.29462\times 10^{14}$
Root an. cond. $4.13641$
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  + 768·4-s + 6.42e3·7-s − 1.31e4·9-s + 4.34e3·11-s + 3.44e5·16-s + 4.99e5·23-s + 8.43e5·25-s + 4.93e6·28-s − 1.27e6·29-s − 1.00e7·36-s + 7.06e6·37-s − 1.13e7·43-s + 3.33e6·44-s + 1.44e7·49-s + 1.97e7·53-s − 8.42e7·63-s + 1.17e8·64-s − 9.39e6·67-s + 5.39e6·71-s + 2.78e7·77-s + 1.34e8·79-s + 1.00e8·81-s + 3.83e8·92-s − 5.70e7·99-s + 6.47e8·100-s + 9.27e8·107-s + 1.66e8·109-s + ⋯
L(s)  = 1  + 3·4-s + 2.67·7-s − 2·9-s + 0.296·11-s + 21/4·16-s + 1.78·23-s + 2.15·25-s + 8.02·28-s − 1.80·29-s − 6·36-s + 3.77·37-s − 3.33·43-s + 0.890·44-s + 2.50·49-s + 2.49·53-s − 5.34·63-s + 7·64-s − 0.466·67-s + 0.212·71-s + 0.793·77-s + 3.45·79-s + 7/3·81-s + 5.35·92-s − 0.593·99-s + 6.47·100-s + 7.07·107-s + 1.18·109-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(2^{12} \cdot 3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(6.29462\times 10^{14}\)
Root analytic conductor: \(4.13641\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{12} \cdot 3^{12} \cdot 7^{12} ,\ ( \ : [4]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(134.2627760\)
\(L(\frac12)\) \(\approx\) \(134.2627760\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p^{7} T^{2} )^{6} \)
3 \( ( 1 + p^{7} T^{2} )^{6} \)
7 \( 1 - 6420 T + 3825942 p T^{2} - 1397441060 p^{2} T^{3} + 6725425545 p^{5} T^{4} - 807552263880 p^{6} T^{5} + 25057929804 p^{10} T^{6} - 807552263880 p^{14} T^{7} + 6725425545 p^{21} T^{8} - 1397441060 p^{26} T^{9} + 3825942 p^{33} T^{10} - 6420 p^{40} T^{11} + p^{48} T^{12} \)
good5 \( 1 - 843012 T^{2} + 502869048714 T^{4} - 6590893127665588 p^{2} T^{6} + 70430404351651168527 p^{4} T^{8} - \)\(65\!\cdots\!88\)\( p^{6} T^{10} + \)\(86\!\cdots\!64\)\( p^{8} T^{12} - \)\(65\!\cdots\!88\)\( p^{22} T^{14} + 70430404351651168527 p^{36} T^{16} - 6590893127665588 p^{50} T^{18} + 502869048714 p^{64} T^{20} - 843012 p^{80} T^{22} + p^{96} T^{24} \)
11 \( ( 1 - 2172 T + 389139054 T^{2} - 6121043174220 T^{3} + 124714029697088739 T^{4} - \)\(12\!\cdots\!80\)\( T^{5} + \)\(43\!\cdots\!80\)\( T^{6} - \)\(12\!\cdots\!80\)\( p^{8} T^{7} + 124714029697088739 p^{16} T^{8} - 6121043174220 p^{24} T^{9} + 389139054 p^{32} T^{10} - 2172 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
13 \( 1 - 6496908108 T^{2} + 1600894332213811530 p T^{4} - \)\(43\!\cdots\!28\)\( T^{6} + \)\(66\!\cdots\!19\)\( T^{8} - \)\(77\!\cdots\!36\)\( T^{10} + \)\(71\!\cdots\!28\)\( T^{12} - \)\(77\!\cdots\!36\)\( p^{16} T^{14} + \)\(66\!\cdots\!19\)\( p^{32} T^{16} - \)\(43\!\cdots\!28\)\( p^{48} T^{18} + 1600894332213811530 p^{65} T^{20} - 6496908108 p^{80} T^{22} + p^{96} T^{24} \)
17 \( 1 - 34075744164 T^{2} + \)\(56\!\cdots\!74\)\( T^{4} - \)\(64\!\cdots\!16\)\( T^{6} + \)\(60\!\cdots\!87\)\( T^{8} - \)\(50\!\cdots\!04\)\( T^{10} + \)\(37\!\cdots\!24\)\( T^{12} - \)\(50\!\cdots\!04\)\( p^{16} T^{14} + \)\(60\!\cdots\!87\)\( p^{32} T^{16} - \)\(64\!\cdots\!16\)\( p^{48} T^{18} + \)\(56\!\cdots\!74\)\( p^{64} T^{20} - 34075744164 p^{80} T^{22} + p^{96} T^{24} \)
19 \( 1 - 152019244284 T^{2} + \)\(11\!\cdots\!50\)\( T^{4} - \)\(53\!\cdots\!04\)\( T^{6} + \)\(17\!\cdots\!91\)\( T^{8} - \)\(44\!\cdots\!64\)\( T^{10} + \)\(86\!\cdots\!36\)\( T^{12} - \)\(44\!\cdots\!64\)\( p^{16} T^{14} + \)\(17\!\cdots\!91\)\( p^{32} T^{16} - \)\(53\!\cdots\!04\)\( p^{48} T^{18} + \)\(11\!\cdots\!50\)\( p^{64} T^{20} - 152019244284 p^{80} T^{22} + p^{96} T^{24} \)
23 \( ( 1 - 249900 T + 150489966030 T^{2} - 59507337243320220 T^{3} + \)\(19\!\cdots\!75\)\( T^{4} - \)\(50\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!00\)\( T^{6} - \)\(50\!\cdots\!40\)\( p^{8} T^{7} + \)\(19\!\cdots\!75\)\( p^{16} T^{8} - 59507337243320220 p^{24} T^{9} + 150489966030 p^{32} T^{10} - 249900 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
29 \( ( 1 + 639204 T + 1418925640578 T^{2} + 782309009443026900 T^{3} + \)\(12\!\cdots\!43\)\( T^{4} + \)\(62\!\cdots\!80\)\( T^{5} + \)\(75\!\cdots\!32\)\( T^{6} + \)\(62\!\cdots\!80\)\( p^{8} T^{7} + \)\(12\!\cdots\!43\)\( p^{16} T^{8} + 782309009443026900 p^{24} T^{9} + 1418925640578 p^{32} T^{10} + 639204 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
31 \( 1 - 5580059030220 T^{2} + \)\(15\!\cdots\!94\)\( T^{4} - \)\(27\!\cdots\!96\)\( T^{6} + \)\(37\!\cdots\!23\)\( T^{8} - \)\(39\!\cdots\!16\)\( T^{10} + \)\(36\!\cdots\!12\)\( T^{12} - \)\(39\!\cdots\!16\)\( p^{16} T^{14} + \)\(37\!\cdots\!23\)\( p^{32} T^{16} - \)\(27\!\cdots\!96\)\( p^{48} T^{18} + \)\(15\!\cdots\!94\)\( p^{64} T^{20} - 5580059030220 p^{80} T^{22} + p^{96} T^{24} \)
37 \( ( 1 - 3534324 T + 12386048168874 T^{2} - 15555030977131161476 T^{3} + \)\(25\!\cdots\!47\)\( T^{4} + \)\(14\!\cdots\!16\)\( T^{5} + \)\(63\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!16\)\( p^{8} T^{7} + \)\(25\!\cdots\!47\)\( p^{16} T^{8} - 15555030977131161476 p^{24} T^{9} + 12386048168874 p^{32} T^{10} - 3534324 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
41 \( 1 - 77931345654756 T^{2} + \)\(28\!\cdots\!22\)\( T^{4} - \)\(66\!\cdots\!44\)\( T^{6} + \)\(10\!\cdots\!91\)\( T^{8} - \)\(13\!\cdots\!36\)\( T^{10} + \)\(11\!\cdots\!68\)\( T^{12} - \)\(13\!\cdots\!36\)\( p^{16} T^{14} + \)\(10\!\cdots\!91\)\( p^{32} T^{16} - \)\(66\!\cdots\!44\)\( p^{48} T^{18} + \)\(28\!\cdots\!22\)\( p^{64} T^{20} - 77931345654756 p^{80} T^{22} + p^{96} T^{24} \)
43 \( ( 1 + 5694012 T + 58707752512122 T^{2} + \)\(28\!\cdots\!16\)\( T^{3} + \)\(16\!\cdots\!19\)\( T^{4} + \)\(59\!\cdots\!20\)\( T^{5} + \)\(24\!\cdots\!64\)\( T^{6} + \)\(59\!\cdots\!20\)\( p^{8} T^{7} + \)\(16\!\cdots\!19\)\( p^{16} T^{8} + \)\(28\!\cdots\!16\)\( p^{24} T^{9} + 58707752512122 p^{32} T^{10} + 5694012 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
47 \( 1 - 212748761135052 T^{2} + \)\(21\!\cdots\!14\)\( T^{4} - \)\(14\!\cdots\!48\)\( T^{6} + \)\(66\!\cdots\!87\)\( T^{8} - \)\(23\!\cdots\!32\)\( T^{10} + \)\(63\!\cdots\!44\)\( T^{12} - \)\(23\!\cdots\!32\)\( p^{16} T^{14} + \)\(66\!\cdots\!87\)\( p^{32} T^{16} - \)\(14\!\cdots\!48\)\( p^{48} T^{18} + \)\(21\!\cdots\!14\)\( p^{64} T^{20} - 212748761135052 p^{80} T^{22} + p^{96} T^{24} \)
53 \( ( 1 - 9857484 T + 110693736580290 T^{2} - \)\(70\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!43\)\( T^{4} - \)\(67\!\cdots\!12\)\( T^{5} + \)\(70\!\cdots\!72\)\( T^{6} - \)\(67\!\cdots\!12\)\( p^{8} T^{7} + \)\(11\!\cdots\!43\)\( p^{16} T^{8} - \)\(70\!\cdots\!92\)\( p^{24} T^{9} + 110693736580290 p^{32} T^{10} - 9857484 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
59 \( 1 - 669802921181676 T^{2} + \)\(21\!\cdots\!18\)\( T^{4} - \)\(46\!\cdots\!60\)\( T^{6} + \)\(72\!\cdots\!23\)\( T^{8} - \)\(16\!\cdots\!00\)\( p T^{10} + \)\(13\!\cdots\!12\)\( T^{12} - \)\(16\!\cdots\!00\)\( p^{17} T^{14} + \)\(72\!\cdots\!23\)\( p^{32} T^{16} - \)\(46\!\cdots\!60\)\( p^{48} T^{18} + \)\(21\!\cdots\!18\)\( p^{64} T^{20} - 669802921181676 p^{80} T^{22} + p^{96} T^{24} \)
61 \( 1 - 911618833195500 T^{2} + \)\(48\!\cdots\!70\)\( T^{4} - \)\(18\!\cdots\!40\)\( T^{6} + \)\(54\!\cdots\!27\)\( T^{8} - \)\(13\!\cdots\!64\)\( T^{10} + \)\(27\!\cdots\!48\)\( T^{12} - \)\(13\!\cdots\!64\)\( p^{16} T^{14} + \)\(54\!\cdots\!27\)\( p^{32} T^{16} - \)\(18\!\cdots\!40\)\( p^{48} T^{18} + \)\(48\!\cdots\!70\)\( p^{64} T^{20} - 911618833195500 p^{80} T^{22} + p^{96} T^{24} \)
67 \( ( 1 + 4697004 T + 1000196801376810 T^{2} - \)\(71\!\cdots\!08\)\( T^{3} + \)\(31\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!68\)\( T^{5} + \)\(58\!\cdots\!92\)\( T^{6} - \)\(12\!\cdots\!68\)\( p^{8} T^{7} + \)\(31\!\cdots\!23\)\( p^{16} T^{8} - \)\(71\!\cdots\!08\)\( p^{24} T^{9} + 1000196801376810 p^{32} T^{10} + 4697004 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
71 \( ( 1 - 2696604 T + 1930841582111502 T^{2} + \)\(18\!\cdots\!84\)\( T^{3} + \)\(22\!\cdots\!11\)\( T^{4} + \)\(24\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!08\)\( T^{6} + \)\(24\!\cdots\!16\)\( p^{8} T^{7} + \)\(22\!\cdots\!11\)\( p^{16} T^{8} + \)\(18\!\cdots\!84\)\( p^{24} T^{9} + 1930841582111502 p^{32} T^{10} - 2696604 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
73 \( 1 - 4400415377930220 T^{2} + \)\(10\!\cdots\!14\)\( T^{4} - \)\(18\!\cdots\!76\)\( T^{6} + \)\(23\!\cdots\!03\)\( T^{8} - \)\(25\!\cdots\!96\)\( T^{10} + \)\(22\!\cdots\!12\)\( T^{12} - \)\(25\!\cdots\!96\)\( p^{16} T^{14} + \)\(23\!\cdots\!03\)\( p^{32} T^{16} - \)\(18\!\cdots\!76\)\( p^{48} T^{18} + \)\(10\!\cdots\!14\)\( p^{64} T^{20} - 4400415377930220 p^{80} T^{22} + p^{96} T^{24} \)
79 \( ( 1 - 67280484 T + 5142148974857802 T^{2} - \)\(25\!\cdots\!36\)\( T^{3} + \)\(14\!\cdots\!71\)\( T^{4} - \)\(62\!\cdots\!88\)\( T^{5} + \)\(26\!\cdots\!04\)\( T^{6} - \)\(62\!\cdots\!88\)\( p^{8} T^{7} + \)\(14\!\cdots\!71\)\( p^{16} T^{8} - \)\(25\!\cdots\!36\)\( p^{24} T^{9} + 5142148974857802 p^{32} T^{10} - 67280484 p^{40} T^{11} + p^{48} T^{12} )^{2} \)
83 \( 1 - 10082227457901996 T^{2} + \)\(61\!\cdots\!34\)\( T^{4} - \)\(26\!\cdots\!88\)\( T^{6} + \)\(93\!\cdots\!55\)\( T^{8} - \)\(27\!\cdots\!76\)\( T^{10} + \)\(66\!\cdots\!56\)\( T^{12} - \)\(27\!\cdots\!76\)\( p^{16} T^{14} + \)\(93\!\cdots\!55\)\( p^{32} T^{16} - \)\(26\!\cdots\!88\)\( p^{48} T^{18} + \)\(61\!\cdots\!34\)\( p^{64} T^{20} - 10082227457901996 p^{80} T^{22} + p^{96} T^{24} \)
89 \( 1 - 18424694009365284 T^{2} + \)\(18\!\cdots\!26\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{6} + \)\(88\!\cdots\!27\)\( T^{8} - \)\(44\!\cdots\!20\)\( T^{10} + \)\(19\!\cdots\!68\)\( T^{12} - \)\(44\!\cdots\!20\)\( p^{16} T^{14} + \)\(88\!\cdots\!27\)\( p^{32} T^{16} - \)\(14\!\cdots\!56\)\( p^{48} T^{18} + \)\(18\!\cdots\!26\)\( p^{64} T^{20} - 18424694009365284 p^{80} T^{22} + p^{96} T^{24} \)
97 \( 1 - 31655639893060716 T^{2} + \)\(43\!\cdots\!70\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(46\!\cdots\!11\)\( T^{8} - \)\(41\!\cdots\!80\)\( T^{10} + \)\(32\!\cdots\!44\)\( T^{12} - \)\(41\!\cdots\!80\)\( p^{16} T^{14} + \)\(46\!\cdots\!11\)\( p^{32} T^{16} - \)\(44\!\cdots\!20\)\( p^{48} T^{18} + \)\(43\!\cdots\!70\)\( p^{64} T^{20} - 31655639893060716 p^{80} T^{22} + p^{96} T^{24} \)
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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

−4.06607739536798412574103094481, −3.80055738182478177460705314423, −3.65828900339240696574384189432, −3.62533851415124913919625734483, −3.43987870296509857487355709143, −3.34917538726892715585274155116, −3.06922983246473525041102474792, −2.94570104527084503521837309197, −2.88921814823130538603852615064, −2.67993617761082142951012198931, −2.52919202354130997095207281235, −2.27779464612023638389007774012, −2.17485458740516278837160594267, −2.17215829715778267067458054514, −1.95005557045804462401331159080, −1.83360804161583418563423225094, −1.51066127663975065163057466393, −1.42292530053344634865579922897, −1.25575346677264443652543554661, −1.13779282354800438393681913358, −0.852176994723553926496144561994, −0.76024769353175859347249378113, −0.59626004992803533870778399820, −0.42634690006181494783743781157, −0.26841445535521899606169678880, 0.26841445535521899606169678880, 0.42634690006181494783743781157, 0.59626004992803533870778399820, 0.76024769353175859347249378113, 0.852176994723553926496144561994, 1.13779282354800438393681913358, 1.25575346677264443652543554661, 1.42292530053344634865579922897, 1.51066127663975065163057466393, 1.83360804161583418563423225094, 1.95005557045804462401331159080, 2.17215829715778267067458054514, 2.17485458740516278837160594267, 2.27779464612023638389007774012, 2.52919202354130997095207281235, 2.67993617761082142951012198931, 2.88921814823130538603852615064, 2.94570104527084503521837309197, 3.06922983246473525041102474792, 3.34917538726892715585274155116, 3.43987870296509857487355709143, 3.62533851415124913919625734483, 3.65828900339240696574384189432, 3.80055738182478177460705314423, 4.06607739536798412574103094481

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

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