Properties

Label 24-688e12-1.1-c4e12-0-0
Degree $24$
Conductor $1.125\times 10^{34}$
Sign $1$
Analytic cond. $1.67414\times 10^{22}$
Root an. cond. $8.43318$
Motivic weight $4$
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  + 255·9-s + 180·11-s − 216·13-s + 678·17-s − 1.56e3·23-s + 3.66e3·25-s − 5.71e3·31-s + 4.87e3·41-s + 1.10e3·43-s + 5.52e3·47-s + 1.01e4·49-s + 1.21e3·53-s − 1.40e4·59-s + 1.08e3·67-s − 2.43e4·79-s + 1.72e4·81-s + 7.03e3·83-s − 5.84e3·97-s + 4.59e4·99-s − 2.52e4·101-s + 2.79e4·103-s + 4.10e3·107-s + 1.15e4·109-s − 5.50e4·117-s − 1.14e5·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 3.14·9-s + 1.48·11-s − 1.27·13-s + 2.34·17-s − 2.96·23-s + 5.86·25-s − 5.94·31-s + 2.90·41-s + 0.599·43-s + 2.50·47-s + 4.22·49-s + 0.431·53-s − 4.02·59-s + 0.242·67-s − 3.89·79-s + 2.62·81-s + 1.02·83-s − 0.620·97-s + 4.68·99-s − 2.47·101-s + 2.63·103-s + 0.358·107-s + 0.975·109-s − 4.02·117-s − 7.85·121-s + 6.20e−5·127-s + 5.82e−5·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s+2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 43^{12}\)
Sign: $1$
Analytic conductor: \(1.67414\times 10^{22}\)
Root analytic conductor: \(8.43318\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 43^{12} ,\ ( \ : [2]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.3381726170\)
\(L(\frac12)\) \(\approx\) \(0.3381726170\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 - 1108 T + 44122 p T^{2} + 2531708 p^{2} T^{3} + 8565509 p^{3} T^{4} + 247619960 p^{4} T^{5} + 10220444708 p^{6} T^{6} + 247619960 p^{8} T^{7} + 8565509 p^{11} T^{8} + 2531708 p^{14} T^{9} + 44122 p^{17} T^{10} - 1108 p^{20} T^{11} + p^{24} T^{12} \)
good3 \( 1 - 85 p T^{2} + 47783 T^{4} - 6646421 T^{6} + 88887767 p^{2} T^{8} - 984092996 p^{4} T^{10} + 9589620802 p^{6} T^{12} - 984092996 p^{12} T^{14} + 88887767 p^{18} T^{16} - 6646421 p^{24} T^{18} + 47783 p^{32} T^{20} - 85 p^{41} T^{22} + p^{48} T^{24} \)
5 \( 1 - 3663 T^{2} + 1302203 p T^{4} - 1473032153 p T^{6} + 5973224140031 T^{8} - 3902236317085676 T^{10} + 2399998678381934162 T^{12} - 3902236317085676 p^{8} T^{14} + 5973224140031 p^{16} T^{16} - 1473032153 p^{25} T^{18} + 1302203 p^{33} T^{20} - 3663 p^{40} T^{22} + p^{48} T^{24} \)
7 \( 1 - 10134 T^{2} + 60060006 T^{4} - 249160434566 T^{6} + 812915964019071 T^{8} - 2246049549704995236 T^{10} + \)\(56\!\cdots\!48\)\( T^{12} - 2246049549704995236 p^{8} T^{14} + 812915964019071 p^{16} T^{16} - 249160434566 p^{24} T^{18} + 60060006 p^{32} T^{20} - 10134 p^{40} T^{22} + p^{48} T^{24} \)
11 \( ( 1 - 90 T + 69626 T^{2} - 4275732 T^{3} + 2084651350 T^{4} - 8260523922 p T^{5} + 309663152918 p^{2} T^{6} - 8260523922 p^{5} T^{7} + 2084651350 p^{8} T^{8} - 4275732 p^{12} T^{9} + 69626 p^{16} T^{10} - 90 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
13 \( ( 1 + 108 T + 6010 p T^{2} + 14124934 T^{3} + 3639591614 T^{4} + 705010114760 T^{5} + 121575713991590 T^{6} + 705010114760 p^{4} T^{7} + 3639591614 p^{8} T^{8} + 14124934 p^{12} T^{9} + 6010 p^{17} T^{10} + 108 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
17 \( ( 1 - 339 T + 345843 T^{2} - 6101049 p T^{3} + 57770879546 T^{4} - 15179725016055 T^{5} + 6016834256618483 T^{6} - 15179725016055 p^{4} T^{7} + 57770879546 p^{8} T^{8} - 6101049 p^{13} T^{9} + 345843 p^{16} T^{10} - 339 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
19 \( 1 - 1175657 T^{2} + 671010023409 T^{4} - 244979577528319381 T^{6} + \)\(63\!\cdots\!63\)\( T^{8} - \)\(12\!\cdots\!10\)\( T^{10} + \)\(18\!\cdots\!98\)\( T^{12} - \)\(12\!\cdots\!10\)\( p^{8} T^{14} + \)\(63\!\cdots\!63\)\( p^{16} T^{16} - 244979577528319381 p^{24} T^{18} + 671010023409 p^{32} T^{20} - 1175657 p^{40} T^{22} + p^{48} T^{24} \)
23 \( ( 1 + 783 T + 1163077 T^{2} + 843582057 T^{3} + 713091466102 T^{4} + 397049506101027 T^{5} + 259574077696956889 T^{6} + 397049506101027 p^{4} T^{7} + 713091466102 p^{8} T^{8} + 843582057 p^{12} T^{9} + 1163077 p^{16} T^{10} + 783 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
29 \( 1 - 5231219 T^{2} + 468703896099 p T^{4} - 23367533349246758977 T^{6} + \)\(29\!\cdots\!35\)\( T^{8} - \)\(29\!\cdots\!88\)\( T^{10} + \)\(23\!\cdots\!82\)\( T^{12} - \)\(29\!\cdots\!88\)\( p^{8} T^{14} + \)\(29\!\cdots\!35\)\( p^{16} T^{16} - 23367533349246758977 p^{24} T^{18} + 468703896099 p^{33} T^{20} - 5231219 p^{40} T^{22} + p^{48} T^{24} \)
31 \( ( 1 + 2855 T + 6518969 T^{2} + 8600011177 T^{3} + 10228398490358 T^{4} + 8682474860253963 T^{5} + 8850492205571119245 T^{6} + 8682474860253963 p^{4} T^{7} + 10228398490358 p^{8} T^{8} + 8600011177 p^{12} T^{9} + 6518969 p^{16} T^{10} + 2855 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
37 \( 1 - 6978249 T^{2} + 25643197308697 T^{4} - 70260487375765593853 T^{6} + \)\(17\!\cdots\!59\)\( T^{8} - \)\(39\!\cdots\!70\)\( T^{10} + \)\(80\!\cdots\!70\)\( T^{12} - \)\(39\!\cdots\!70\)\( p^{8} T^{14} + \)\(17\!\cdots\!59\)\( p^{16} T^{16} - 70260487375765593853 p^{24} T^{18} + 25643197308697 p^{32} T^{20} - 6978249 p^{40} T^{22} + p^{48} T^{24} \)
41 \( ( 1 - 2439 T + 13822603 T^{2} - 19943096547 T^{3} + 69051390319336 T^{4} - 65595635499999207 T^{5} + \)\(21\!\cdots\!21\)\( T^{6} - 65595635499999207 p^{4} T^{7} + 69051390319336 p^{8} T^{8} - 19943096547 p^{12} T^{9} + 13822603 p^{16} T^{10} - 2439 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
47 \( ( 1 - 2763 T + 21754149 T^{2} - 55851382377 T^{3} + 234662825346711 T^{4} - 481604898621884868 T^{5} + \)\(14\!\cdots\!34\)\( T^{6} - 481604898621884868 p^{4} T^{7} + 234662825346711 p^{8} T^{8} - 55851382377 p^{12} T^{9} + 21754149 p^{16} T^{10} - 2763 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
53 \( ( 1 - 606 T + 25718024 T^{2} - 24674633094 T^{3} + 346581142619392 T^{4} - 336085601563787574 T^{5} + \)\(32\!\cdots\!82\)\( T^{6} - 336085601563787574 p^{4} T^{7} + 346581142619392 p^{8} T^{8} - 24674633094 p^{12} T^{9} + 25718024 p^{16} T^{10} - 606 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
59 \( ( 1 + 7008 T + 80137258 T^{2} + 394451068512 T^{3} + 2532059909380847 T^{4} + 9247600606403540352 T^{5} + \)\(41\!\cdots\!80\)\( T^{6} + 9247600606403540352 p^{4} T^{7} + 2532059909380847 p^{8} T^{8} + 394451068512 p^{12} T^{9} + 80137258 p^{16} T^{10} + 7008 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
61 \( 1 - 32665486 T^{2} + 1118892201421590 T^{4} - \)\(24\!\cdots\!14\)\( T^{6} + \)\(52\!\cdots\!07\)\( T^{8} - \)\(83\!\cdots\!88\)\( T^{10} + \)\(13\!\cdots\!08\)\( T^{12} - \)\(83\!\cdots\!88\)\( p^{8} T^{14} + \)\(52\!\cdots\!07\)\( p^{16} T^{16} - \)\(24\!\cdots\!14\)\( p^{24} T^{18} + 1118892201421590 p^{32} T^{20} - 32665486 p^{40} T^{22} + p^{48} T^{24} \)
67 \( ( 1 - 544 T + 66684038 T^{2} - 18315837698 T^{3} + 2452178722840730 T^{4} - 587634295199545788 T^{5} + \)\(60\!\cdots\!70\)\( T^{6} - 587634295199545788 p^{4} T^{7} + 2452178722840730 p^{8} T^{8} - 18315837698 p^{12} T^{9} + 66684038 p^{16} T^{10} - 544 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
71 \( 1 - 101562020 T^{2} + 5784818387848562 T^{4} - \)\(20\!\cdots\!64\)\( T^{6} + \)\(47\!\cdots\!67\)\( T^{8} - \)\(75\!\cdots\!84\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{12} - \)\(75\!\cdots\!84\)\( p^{8} T^{14} + \)\(47\!\cdots\!67\)\( p^{16} T^{16} - \)\(20\!\cdots\!64\)\( p^{24} T^{18} + 5784818387848562 p^{32} T^{20} - 101562020 p^{40} T^{22} + p^{48} T^{24} \)
73 \( 1 - 193491698 T^{2} + 19000747654703030 T^{4} - \)\(12\!\cdots\!18\)\( T^{6} + \)\(60\!\cdots\!75\)\( T^{8} - \)\(23\!\cdots\!56\)\( T^{10} + \)\(72\!\cdots\!44\)\( T^{12} - \)\(23\!\cdots\!56\)\( p^{8} T^{14} + \)\(60\!\cdots\!75\)\( p^{16} T^{16} - \)\(12\!\cdots\!18\)\( p^{24} T^{18} + 19000747654703030 p^{32} T^{20} - 193491698 p^{40} T^{22} + p^{48} T^{24} \)
79 \( ( 1 + 12151 T + 155715789 T^{2} + 10136983391 p T^{3} + 4128429862975343 T^{4} - 5685482156720652576 T^{5} - \)\(30\!\cdots\!42\)\( T^{6} - 5685482156720652576 p^{4} T^{7} + 4128429862975343 p^{8} T^{8} + 10136983391 p^{13} T^{9} + 155715789 p^{16} T^{10} + 12151 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
83 \( ( 1 - 3516 T + 117162268 T^{2} - 863865543048 T^{3} + 7914887456622244 T^{4} - 75598926847021341108 T^{5} + \)\(40\!\cdots\!22\)\( T^{6} - 75598926847021341108 p^{4} T^{7} + 7914887456622244 p^{8} T^{8} - 863865543048 p^{12} T^{9} + 117162268 p^{16} T^{10} - 3516 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
89 \( 1 - 474457834 T^{2} + 1254552724574358 p T^{4} - \)\(17\!\cdots\!66\)\( T^{6} + \)\(19\!\cdots\!11\)\( T^{8} - \)\(17\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!28\)\( T^{12} - \)\(17\!\cdots\!92\)\( p^{8} T^{14} + \)\(19\!\cdots\!11\)\( p^{16} T^{16} - \)\(17\!\cdots\!66\)\( p^{24} T^{18} + 1254552724574358 p^{33} T^{20} - 474457834 p^{40} T^{22} + p^{48} T^{24} \)
97 \( ( 1 + 2921 T + 384124299 T^{2} + 837647682043 T^{3} + 69794294829141482 T^{4} + \)\(11\!\cdots\!33\)\( T^{5} + \)\(76\!\cdots\!95\)\( T^{6} + \)\(11\!\cdots\!33\)\( p^{4} T^{7} + 69794294829141482 p^{8} T^{8} + 837647682043 p^{12} T^{9} + 384124299 p^{16} T^{10} + 2921 p^{20} T^{11} + p^{24} T^{12} )^{2} \)
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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

−2.63182112105232419252219778883, −2.62750910840383320622646273003, −2.59236939581428303492805728274, −2.45705945984204162950417838505, −2.31637729017239491785055490373, −2.24757914800371568924526152474, −2.14947420746335866941160755976, −2.11414086603097433868990836038, −1.81284855264196796104610131574, −1.64391778317133922858775105930, −1.63635963477861341501775179375, −1.48904117870924614759535255360, −1.46549029822792249269872381594, −1.36413412820279608786042519875, −1.30300736645609123049108722513, −1.14019182936605547576603844198, −1.08810476257767881809872177691, −0.944529140039154651847293922508, −0.909445838639947224819395017005, −0.846594512369610318939104590746, −0.55170700119465375561206059233, −0.44113368928918124714758944241, −0.37266631339751214937578187340, −0.098783986261198633154608917457, −0.02775135333818372909710693053, 0.02775135333818372909710693053, 0.098783986261198633154608917457, 0.37266631339751214937578187340, 0.44113368928918124714758944241, 0.55170700119465375561206059233, 0.846594512369610318939104590746, 0.909445838639947224819395017005, 0.944529140039154651847293922508, 1.08810476257767881809872177691, 1.14019182936605547576603844198, 1.30300736645609123049108722513, 1.36413412820279608786042519875, 1.46549029822792249269872381594, 1.48904117870924614759535255360, 1.63635963477861341501775179375, 1.64391778317133922858775105930, 1.81284855264196796104610131574, 2.11414086603097433868990836038, 2.14947420746335866941160755976, 2.24757914800371568924526152474, 2.31637729017239491785055490373, 2.45705945984204162950417838505, 2.59236939581428303492805728274, 2.62750910840383320622646273003, 2.63182112105232419252219778883

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

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