Properties

Degree 24
Conductor $ 43^{12} $
Sign $1$
Motivic weight 4
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 50·4-s + 255·9-s − 180·11-s − 216·13-s + 1.15e3·16-s + 678·17-s + 1.56e3·23-s + 3.66e3·25-s + 5.71e3·31-s + 1.27e4·36-s + 4.87e3·41-s − 1.10e3·43-s − 9.00e3·44-s − 5.52e3·47-s + 1.01e4·49-s − 1.08e4·52-s + 1.21e3·53-s + 1.40e4·59-s + 1.61e4·64-s − 1.08e3·67-s + 3.39e4·68-s + 2.43e4·79-s + 1.72e4·81-s − 7.03e3·83-s + 7.83e4·92-s − 5.84e3·97-s − 4.59e4·99-s + ⋯
L(s)  = 1  + 25/8·4-s + 3.14·9-s − 1.48·11-s − 1.27·13-s + 4.49·16-s + 2.34·17-s + 2.96·23-s + 5.86·25-s + 5.94·31-s + 9.83·36-s + 2.90·41-s − 0.599·43-s − 4.64·44-s − 2.50·47-s + 4.22·49-s − 3.99·52-s + 0.431·53-s + 4.02·59-s + 3.94·64-s − 0.242·67-s + 7.33·68-s + 3.89·79-s + 2.62·81-s − 1.02·83-s + 9.25·92-s − 0.620·97-s − 4.68·99-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(43^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((24,\ 43^{12} ,\ ( \ : [2]^{12} ),\ 1 )\)
\(L(\frac{5}{2})\)  \(\approx\)  \(86.8772\)
\(L(\frac12)\)  \(\approx\)  \(86.8772\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 24. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad43 \( 1 + 1108 T + 44122 p T^{2} - 2531708 p^{2} T^{3} + 8565509 p^{3} T^{4} - 247619960 p^{4} T^{5} + 10220444708 p^{6} T^{6} - 247619960 p^{8} T^{7} + 8565509 p^{11} T^{8} - 2531708 p^{14} T^{9} + 44122 p^{17} T^{10} + 1108 p^{20} T^{11} + p^{24} T^{12} \)
good2 \( 1 - 25 p T^{2} + 1349 T^{4} - 3257 p^{3} T^{6} + 3619 p^{7} T^{8} - 136749 p^{6} T^{10} + 9432207 p^{4} T^{12} - 136749 p^{14} T^{14} + 3619 p^{23} T^{16} - 3257 p^{27} T^{18} + 1349 p^{32} T^{20} - 25 p^{41} T^{22} + p^{48} T^{24} \)
3 \( 1 - 85 p T^{2} + 47783 T^{4} - 6646421 T^{6} + 88887767 p^{2} T^{8} - 984092996 p^{4} T^{10} + 9589620802 p^{6} T^{12} - 984092996 p^{12} T^{14} + 88887767 p^{18} T^{16} - 6646421 p^{24} T^{18} + 47783 p^{32} T^{20} - 85 p^{41} T^{22} + p^{48} T^{24} \)
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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.21180956795999054692394872920, −4.96499534700002864458523298426, −4.86267442932936355387105180397, −4.82798709935685033398652664996, −4.72251665884262132833205219136, −4.60770522893687788492882239185, −4.33869015343380834292318253105, −4.16893775230296432454822732375, −3.96391021400871232148927834228, −3.63133345149936214909234479107, −3.58652799255498187144706696335, −3.26704972758665571293883281250, −3.06927732898238882247780417775, −2.76516038077872148322488566760, −2.62780511990073890195166163342, −2.58297413036754763427615799148, −2.46760618385530248088167326570, −2.46503354508459468744653875224, −2.39272410498991029173531096994, −1.27544788562586107829633175294, −1.26527506817634137306877896819, −1.22828245147751661481197054843, −1.16354562543860040550095235321, −0.878613051298174907659235221261, −0.66081432769580188627961834943, 0.66081432769580188627961834943, 0.878613051298174907659235221261, 1.16354562543860040550095235321, 1.22828245147751661481197054843, 1.26527506817634137306877896819, 1.27544788562586107829633175294, 2.39272410498991029173531096994, 2.46503354508459468744653875224, 2.46760618385530248088167326570, 2.58297413036754763427615799148, 2.62780511990073890195166163342, 2.76516038077872148322488566760, 3.06927732898238882247780417775, 3.26704972758665571293883281250, 3.58652799255498187144706696335, 3.63133345149936214909234479107, 3.96391021400871232148927834228, 4.16893775230296432454822732375, 4.33869015343380834292318253105, 4.60770522893687788492882239185, 4.72251665884262132833205219136, 4.82798709935685033398652664996, 4.86267442932936355387105180397, 4.96499534700002864458523298426, 5.21180956795999054692394872920

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

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