# Properties

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

# Origins of factors

## 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 $$( 1 + 2921 T + 384124299 T^{2} + 837647682043 T^{3} + 69794294829141482 T^{4} +$$$$11\!\cdots\!33$$$$T^{5} +$$$$76\!\cdots\!95$$$$T^{6} +$$$$11\!\cdots\!33$$$$p^{4} T^{7} + 69794294829141482 p^{8} T^{8} + 837647682043 p^{12} T^{9} + 384124299 p^{16} T^{10} + 2921 p^{20} T^{11} + p^{24} T^{12} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}