Properties

Label 24-21e12-1.1-c6e12-0-0
Degree $24$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $1.61655\times 10^{8}$
Root an. cond. $2.19798$
Motivic weight $6$
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  + 52·3-s + 126·4-s + 1.03e3·9-s + 6.55e3·12-s + 384·13-s + 1.15e3·16-s + 1.13e4·19-s + 5.51e4·25-s + 4.49e4·27-s − 9.36e3·31-s + 1.29e5·36-s + 2.12e5·37-s + 1.99e4·39-s + 2.80e4·43-s + 6.00e4·48-s + 1.00e5·49-s + 4.83e4·52-s + 5.87e5·57-s + 9.26e5·61-s − 6.88e5·64-s − 2.23e5·67-s + 6.00e5·73-s + 2.86e6·75-s + 1.42e6·76-s − 2.76e5·79-s + 2.06e6·81-s − 4.86e5·93-s + ⋯
L(s)  = 1  + 1.92·3-s + 1.96·4-s + 1.41·9-s + 3.79·12-s + 0.174·13-s + 0.281·16-s + 1.64·19-s + 3.52·25-s + 2.28·27-s − 0.314·31-s + 2.78·36-s + 4.18·37-s + 0.336·39-s + 0.353·43-s + 0.543·48-s + 6/7·49-s + 0.344·52-s + 3.17·57-s + 4.08·61-s − 2.62·64-s − 0.741·67-s + 1.54·73-s + 6.79·75-s + 3.24·76-s − 0.561·79-s + 3.88·81-s − 0.605·93-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.61655\times 10^{8}\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 7^{12} ,\ ( \ : [3]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(68.68635815\)
\(L(\frac12)\) \(\approx\) \(68.68635815\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 52 T + 62 p^{3} T^{2} - 8716 p^{2} T^{3} + 32435 p^{4} T^{4} - 940480 p^{4} T^{5} + 1056172 p^{7} T^{6} - 940480 p^{10} T^{7} + 32435 p^{16} T^{8} - 8716 p^{20} T^{9} + 62 p^{27} T^{10} - 52 p^{30} T^{11} + p^{36} T^{12} \)
7 \( ( 1 - p^{5} T^{2} )^{6} \)
good2 \( 1 - 63 p T^{2} + 14721 T^{4} - 255221 p^{2} T^{6} + 19824021 p^{2} T^{8} - 7513401 p^{9} T^{10} + 16704875 p^{14} T^{12} - 7513401 p^{21} T^{14} + 19824021 p^{26} T^{16} - 255221 p^{38} T^{18} + 14721 p^{48} T^{20} - 63 p^{61} T^{22} + p^{72} T^{24} \)
5 \( 1 - 55104 T^{2} + 1468600854 T^{4} - 1121223216128 p^{2} T^{6} + 715450128353247 p^{4} T^{8} - 446836833704952768 p^{6} T^{10} + \)\(28\!\cdots\!56\)\( p^{8} T^{12} - 446836833704952768 p^{18} T^{14} + 715450128353247 p^{28} T^{16} - 1121223216128 p^{38} T^{18} + 1468600854 p^{48} T^{20} - 55104 p^{60} T^{22} + p^{72} T^{24} \)
11 \( 1 - 11693640 T^{2} + 60395638769214 T^{4} - \)\(18\!\cdots\!40\)\( T^{6} + \)\(39\!\cdots\!27\)\( T^{8} - \)\(70\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!44\)\( T^{12} - \)\(70\!\cdots\!92\)\( p^{12} T^{14} + \)\(39\!\cdots\!27\)\( p^{24} T^{16} - \)\(18\!\cdots\!40\)\( p^{36} T^{18} + 60395638769214 p^{48} T^{20} - 11693640 p^{60} T^{22} + p^{72} T^{24} \)
13 \( ( 1 - 192 T + 19221162 T^{2} + 1668836808 T^{3} + 177653178388515 T^{4} + 39139093191394344 T^{5} + \)\(10\!\cdots\!56\)\( T^{6} + 39139093191394344 p^{6} T^{7} + 177653178388515 p^{12} T^{8} + 1668836808 p^{18} T^{9} + 19221162 p^{24} T^{10} - 192 p^{30} T^{11} + p^{36} T^{12} )^{2} \)
17 \( 1 - 124265232 T^{2} + 8075396374929198 T^{4} - \)\(35\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!87\)\( T^{8} - \)\(33\!\cdots\!00\)\( T^{10} + \)\(84\!\cdots\!68\)\( T^{12} - \)\(33\!\cdots\!00\)\( p^{12} T^{14} + \)\(12\!\cdots\!87\)\( p^{24} T^{16} - \)\(35\!\cdots\!40\)\( p^{36} T^{18} + 8075396374929198 p^{48} T^{20} - 124265232 p^{60} T^{22} + p^{72} T^{24} \)
19 \( ( 1 - 5652 T + 179972934 T^{2} - 897430944636 T^{3} + 14455190511039315 T^{4} - 65727793020462160176 T^{5} + \)\(40\!\cdots\!68\)\( p T^{6} - 65727793020462160176 p^{6} T^{7} + 14455190511039315 p^{12} T^{8} - 897430944636 p^{18} T^{9} + 179972934 p^{24} T^{10} - 5652 p^{30} T^{11} + p^{36} T^{12} )^{2} \)
23 \( 1 - 928267704 T^{2} + 456924799440799326 T^{4} - \)\(15\!\cdots\!16\)\( T^{6} + \)\(39\!\cdots\!99\)\( T^{8} - \)\(80\!\cdots\!64\)\( T^{10} + \)\(13\!\cdots\!96\)\( T^{12} - \)\(80\!\cdots\!64\)\( p^{12} T^{14} + \)\(39\!\cdots\!99\)\( p^{24} T^{16} - \)\(15\!\cdots\!16\)\( p^{36} T^{18} + 456924799440799326 p^{48} T^{20} - 928267704 p^{60} T^{22} + p^{72} T^{24} \)
29 \( 1 - 3560516076 T^{2} + 6412459540016584386 T^{4} - \)\(76\!\cdots\!20\)\( T^{6} + \)\(23\!\cdots\!87\)\( p T^{8} - \)\(49\!\cdots\!92\)\( T^{10} + \)\(30\!\cdots\!60\)\( T^{12} - \)\(49\!\cdots\!92\)\( p^{12} T^{14} + \)\(23\!\cdots\!87\)\( p^{25} T^{16} - \)\(76\!\cdots\!20\)\( p^{36} T^{18} + 6412459540016584386 p^{48} T^{20} - 3560516076 p^{60} T^{22} + p^{72} T^{24} \)
31 \( ( 1 + 4680 T + 3236254218 T^{2} + 31562116597576 T^{3} + 5286868070613701343 T^{4} + \)\(60\!\cdots\!04\)\( T^{5} + \)\(56\!\cdots\!84\)\( T^{6} + \)\(60\!\cdots\!04\)\( p^{6} T^{7} + 5286868070613701343 p^{12} T^{8} + 31562116597576 p^{18} T^{9} + 3236254218 p^{24} T^{10} + 4680 p^{30} T^{11} + p^{36} T^{12} )^{2} \)
37 \( ( 1 - 106008 T + 17251806222 T^{2} - 34202925512408 p T^{3} + \)\(11\!\cdots\!03\)\( T^{4} - \)\(62\!\cdots\!84\)\( T^{5} + \)\(39\!\cdots\!16\)\( T^{6} - \)\(62\!\cdots\!84\)\( p^{6} T^{7} + \)\(11\!\cdots\!03\)\( p^{12} T^{8} - 34202925512408 p^{19} T^{9} + 17251806222 p^{24} T^{10} - 106008 p^{30} T^{11} + p^{36} T^{12} )^{2} \)
41 \( 1 - 39047495184 T^{2} + \)\(73\!\cdots\!82\)\( T^{4} - \)\(87\!\cdots\!04\)\( T^{6} + \)\(75\!\cdots\!87\)\( T^{8} - \)\(50\!\cdots\!16\)\( T^{10} + \)\(26\!\cdots\!64\)\( T^{12} - \)\(50\!\cdots\!16\)\( p^{12} T^{14} + \)\(75\!\cdots\!87\)\( p^{24} T^{16} - \)\(87\!\cdots\!04\)\( p^{36} T^{18} + \)\(73\!\cdots\!82\)\( p^{48} T^{20} - 39047495184 p^{60} T^{22} + p^{72} T^{24} \)
43 \( ( 1 - 14040 T + 9749002014 T^{2} + 303502458318728 T^{3} + 83456374862616461487 T^{4} + \)\(11\!\cdots\!20\)\( T^{5} + \)\(68\!\cdots\!44\)\( T^{6} + \)\(11\!\cdots\!20\)\( p^{6} T^{7} + 83456374862616461487 p^{12} T^{8} + 303502458318728 p^{18} T^{9} + 9749002014 p^{24} T^{10} - 14040 p^{30} T^{11} + p^{36} T^{12} )^{2} \)
47 \( 1 - 49641718140 T^{2} + \)\(11\!\cdots\!42\)\( T^{4} - \)\(14\!\cdots\!84\)\( T^{6} + \)\(93\!\cdots\!71\)\( T^{8} + \)\(28\!\cdots\!04\)\( T^{10} - \)\(10\!\cdots\!44\)\( T^{12} + \)\(28\!\cdots\!04\)\( p^{12} T^{14} + \)\(93\!\cdots\!71\)\( p^{24} T^{16} - \)\(14\!\cdots\!84\)\( p^{36} T^{18} + \)\(11\!\cdots\!42\)\( p^{48} T^{20} - 49641718140 p^{60} T^{22} + p^{72} T^{24} \)
53 \( 1 - 161105116284 T^{2} + \)\(12\!\cdots\!30\)\( T^{4} - \)\(58\!\cdots\!04\)\( T^{6} + \)\(19\!\cdots\!35\)\( T^{8} - \)\(54\!\cdots\!32\)\( T^{10} + \)\(12\!\cdots\!52\)\( T^{12} - \)\(54\!\cdots\!32\)\( p^{12} T^{14} + \)\(19\!\cdots\!35\)\( p^{24} T^{16} - \)\(58\!\cdots\!04\)\( p^{36} T^{18} + \)\(12\!\cdots\!30\)\( p^{48} T^{20} - 161105116284 p^{60} T^{22} + p^{72} T^{24} \)
59 \( 1 - 337822020396 T^{2} + \)\(56\!\cdots\!70\)\( T^{4} - \)\(62\!\cdots\!48\)\( T^{6} + \)\(50\!\cdots\!23\)\( T^{8} - \)\(30\!\cdots\!20\)\( T^{10} + \)\(14\!\cdots\!04\)\( T^{12} - \)\(30\!\cdots\!20\)\( p^{12} T^{14} + \)\(50\!\cdots\!23\)\( p^{24} T^{16} - \)\(62\!\cdots\!48\)\( p^{36} T^{18} + \)\(56\!\cdots\!70\)\( p^{48} T^{20} - 337822020396 p^{60} T^{22} + p^{72} T^{24} \)
61 \( ( 1 - 463368 T + 274969916586 T^{2} - 100295001713623824 T^{3} + \)\(34\!\cdots\!15\)\( T^{4} - \)\(93\!\cdots\!16\)\( T^{5} + \)\(23\!\cdots\!24\)\( T^{6} - \)\(93\!\cdots\!16\)\( p^{6} T^{7} + \)\(34\!\cdots\!15\)\( p^{12} T^{8} - 100295001713623824 p^{18} T^{9} + 274969916586 p^{24} T^{10} - 463368 p^{30} T^{11} + p^{36} T^{12} )^{2} \)
67 \( ( 1 + 111552 T + 143020276686 T^{2} - 3395120375363616 T^{3} + \)\(81\!\cdots\!19\)\( T^{4} - \)\(14\!\cdots\!68\)\( T^{5} + \)\(62\!\cdots\!12\)\( T^{6} - \)\(14\!\cdots\!68\)\( p^{6} T^{7} + \)\(81\!\cdots\!19\)\( p^{12} T^{8} - 3395120375363616 p^{18} T^{9} + 143020276686 p^{24} T^{10} + 111552 p^{30} T^{11} + p^{36} T^{12} )^{2} \)
71 \( 1 - 401361224808 T^{2} + \)\(12\!\cdots\!18\)\( T^{4} - \)\(27\!\cdots\!04\)\( T^{6} + \)\(51\!\cdots\!67\)\( T^{8} - \)\(79\!\cdots\!76\)\( T^{10} + \)\(11\!\cdots\!08\)\( T^{12} - \)\(79\!\cdots\!76\)\( p^{12} T^{14} + \)\(51\!\cdots\!67\)\( p^{24} T^{16} - \)\(27\!\cdots\!04\)\( p^{36} T^{18} + \)\(12\!\cdots\!18\)\( p^{48} T^{20} - 401361224808 p^{60} T^{22} + p^{72} T^{24} \)
73 \( ( 1 - 300492 T + 872431102194 T^{2} - 211773247621274012 T^{3} + \)\(32\!\cdots\!79\)\( T^{4} - \)\(61\!\cdots\!24\)\( T^{5} + \)\(64\!\cdots\!00\)\( T^{6} - \)\(61\!\cdots\!24\)\( p^{6} T^{7} + \)\(32\!\cdots\!79\)\( p^{12} T^{8} - 211773247621274012 p^{18} T^{9} + 872431102194 p^{24} T^{10} - 300492 p^{30} T^{11} + p^{36} T^{12} )^{2} \)
79 \( ( 1 + 138432 T + 745227854070 T^{2} + 239274317465345984 T^{3} + \)\(28\!\cdots\!87\)\( T^{4} + \)\(12\!\cdots\!72\)\( T^{5} + \)\(74\!\cdots\!12\)\( T^{6} + \)\(12\!\cdots\!72\)\( p^{6} T^{7} + \)\(28\!\cdots\!87\)\( p^{12} T^{8} + 239274317465345984 p^{18} T^{9} + 745227854070 p^{24} T^{10} + 138432 p^{30} T^{11} + p^{36} T^{12} )^{2} \)
83 \( 1 - 2422223724828 T^{2} + \)\(25\!\cdots\!02\)\( T^{4} - \)\(15\!\cdots\!28\)\( T^{6} + \)\(59\!\cdots\!39\)\( T^{8} - \)\(15\!\cdots\!60\)\( T^{10} + \)\(38\!\cdots\!20\)\( T^{12} - \)\(15\!\cdots\!60\)\( p^{12} T^{14} + \)\(59\!\cdots\!39\)\( p^{24} T^{16} - \)\(15\!\cdots\!28\)\( p^{36} T^{18} + \)\(25\!\cdots\!02\)\( p^{48} T^{20} - 2422223724828 p^{60} T^{22} + p^{72} T^{24} \)
89 \( 1 - 2000094606384 T^{2} + \)\(28\!\cdots\!10\)\( T^{4} - \)\(28\!\cdots\!32\)\( T^{6} + \)\(22\!\cdots\!11\)\( T^{8} - \)\(14\!\cdots\!44\)\( T^{10} + \)\(81\!\cdots\!52\)\( T^{12} - \)\(14\!\cdots\!44\)\( p^{12} T^{14} + \)\(22\!\cdots\!11\)\( p^{24} T^{16} - \)\(28\!\cdots\!32\)\( p^{36} T^{18} + \)\(28\!\cdots\!10\)\( p^{48} T^{20} - 2000094606384 p^{60} T^{22} + p^{72} T^{24} \)
97 \( ( 1 - 810708 T + 2967260359170 T^{2} - 1871491030983620996 T^{3} + \)\(47\!\cdots\!11\)\( p T^{4} - \)\(24\!\cdots\!60\)\( T^{5} + \)\(46\!\cdots\!16\)\( T^{6} - \)\(24\!\cdots\!60\)\( p^{6} T^{7} + \)\(47\!\cdots\!11\)\( p^{13} T^{8} - 1871491030983620996 p^{18} T^{9} + 2967260359170 p^{24} T^{10} - 810708 p^{30} T^{11} + p^{36} T^{12} )^{2} \)
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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

−5.90122800610158876267459602792, −5.41510531109308055729597557572, −5.35118093214802912043057852106, −5.17751341206966365382958125441, −4.96771715157316624203870195215, −4.84489684295728465501697310484, −4.60098212351708496930235408665, −4.54495670151747176774254494294, −4.17058359341817361858497612376, −3.86643051856995147371921753482, −3.84382803302262106565075646599, −3.61037685467188472779699927690, −3.15508755600966401002801477697, −3.02622517914671722806383721871, −2.99262467366116793143473840113, −2.62505627958476827898364828270, −2.51651143566173053935851017952, −2.44922258492772732693019279014, −2.27323561465292267517968495733, −1.97951011247600285315682796959, −1.33912215542198745202664543877, −1.13113987237378904863084955544, −1.11835804180406885395044860654, −0.59708101761326252455495282298, −0.57204046252275183881874679296, 0.57204046252275183881874679296, 0.59708101761326252455495282298, 1.11835804180406885395044860654, 1.13113987237378904863084955544, 1.33912215542198745202664543877, 1.97951011247600285315682796959, 2.27323561465292267517968495733, 2.44922258492772732693019279014, 2.51651143566173053935851017952, 2.62505627958476827898364828270, 2.99262467366116793143473840113, 3.02622517914671722806383721871, 3.15508755600966401002801477697, 3.61037685467188472779699927690, 3.84382803302262106565075646599, 3.86643051856995147371921753482, 4.17058359341817361858497612376, 4.54495670151747176774254494294, 4.60098212351708496930235408665, 4.84489684295728465501697310484, 4.96771715157316624203870195215, 5.17751341206966365382958125441, 5.35118093214802912043057852106, 5.41510531109308055729597557572, 5.90122800610158876267459602792

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

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