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Properties

Label 24-75e12-1.1-c8e12-0-0
Degree $24$
Conductor $3.168\times 10^{22}$
Sign $1$
Analytic cond. $6.61780\times 10^{17}$
Root an. cond. $5.52751$
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  + 684·4-s + 1.14e4·9-s + 2.28e5·16-s − 2.76e5·19-s − 2.79e5·31-s + 7.80e6·36-s − 3.14e7·49-s − 9.19e7·61-s + 5.74e7·64-s − 1.88e8·76-s − 4.20e8·79-s + 8.97e7·81-s − 2.13e8·109-s + 8.73e8·121-s − 1.90e8·124-s + 127-s + 131-s + 137-s + 139-s + 2.60e9·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6.70e9·169-s − 3.15e9·171-s + ⋯
L(s)  = 1  + 2.67·4-s + 1.73·9-s + 3.48·16-s − 2.11·19-s − 0.302·31-s + 4.64·36-s − 5.46·49-s − 6.64·61-s + 3.42·64-s − 5.66·76-s − 10.7·79-s + 2.08·81-s − 1.51·109-s + 4.07·121-s − 0.807·124-s + 6.05·144-s − 8.21·169-s − 3.68·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+4)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(6.61780\times 10^{17}\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 5^{24} ,\ ( \ : [4]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.654970812\)
\(L(\frac12)\) \(\approx\) \(1.654970812\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1268 p^{2} T^{2} + 18517 p^{7} T^{4} - 770504 p^{10} T^{6} + 18517 p^{23} T^{8} - 1268 p^{34} T^{10} + p^{48} T^{12} \)
5 \( 1 \)
good2 \( ( 1 - 171 p T^{2} + 1917 p^{5} T^{4} - 91187 p^{7} T^{6} + 1917 p^{21} T^{8} - 171 p^{33} T^{10} + p^{48} T^{12} )^{2} \)
7 \( ( 1 + 2248404 p T^{2} + 464565580593 p^{3} T^{4} + 61437958537363592 p^{5} T^{6} + 464565580593 p^{19} T^{8} + 2248404 p^{33} T^{10} + p^{48} T^{12} )^{2} \)
11 \( ( 1 - 436937046 T^{2} + 175639663144304655 T^{4} - \)\(35\!\cdots\!80\)\( p T^{6} + 175639663144304655 p^{16} T^{8} - 436937046 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
13 \( ( 1 + 3352650918 T^{2} + 5571123366751095759 T^{4} + \)\(56\!\cdots\!24\)\( T^{6} + 5571123366751095759 p^{16} T^{8} + 3352650918 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
17 \( ( 1 - 35807249502 T^{2} + \)\(56\!\cdots\!39\)\( T^{4} - \)\(51\!\cdots\!16\)\( T^{6} + \)\(56\!\cdots\!39\)\( p^{16} T^{8} - 35807249502 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
19 \( ( 1 + 69048 T + 23544417879 T^{2} + 2811543295879424 T^{3} + 23544417879 p^{8} T^{4} + 69048 p^{16} T^{5} + p^{24} T^{6} )^{4} \)
23 \( ( 1 - 386927703252 T^{2} + \)\(66\!\cdots\!59\)\( T^{4} - \)\(66\!\cdots\!96\)\( T^{6} + \)\(66\!\cdots\!59\)\( p^{16} T^{8} - 386927703252 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
29 \( ( 1 - 525299162406 T^{2} + \)\(23\!\cdots\!75\)\( p T^{4} - \)\(22\!\cdots\!60\)\( T^{6} + \)\(23\!\cdots\!75\)\( p^{17} T^{8} - 525299162406 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
31 \( ( 1 + 69804 T + 1303500886695 T^{2} + 492707941350291160 T^{3} + 1303500886695 p^{8} T^{4} + 69804 p^{16} T^{5} + p^{24} T^{6} )^{4} \)
37 \( ( 1 + 12451276202118 T^{2} + \)\(83\!\cdots\!59\)\( T^{4} + \)\(36\!\cdots\!24\)\( T^{6} + \)\(83\!\cdots\!59\)\( p^{16} T^{8} + 12451276202118 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
41 \( ( 1 - 46349840377986 T^{2} + \)\(90\!\cdots\!55\)\( T^{4} - \)\(95\!\cdots\!80\)\( T^{6} + \)\(90\!\cdots\!55\)\( p^{16} T^{8} - 46349840377986 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
43 \( ( 1 + 20062514398668 T^{2} - \)\(10\!\cdots\!01\)\( T^{4} - \)\(48\!\cdots\!36\)\( T^{6} - \)\(10\!\cdots\!01\)\( p^{16} T^{8} + 20062514398668 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
47 \( ( 1 - 125469659770932 T^{2} + \)\(69\!\cdots\!59\)\( T^{4} - \)\(21\!\cdots\!16\)\( T^{6} + \)\(69\!\cdots\!59\)\( p^{16} T^{8} - 125469659770932 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
53 \( ( 1 - 195684137175582 T^{2} + \)\(17\!\cdots\!59\)\( T^{4} - \)\(11\!\cdots\!16\)\( T^{6} + \)\(17\!\cdots\!59\)\( p^{16} T^{8} - 195684137175582 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
59 \( ( 1 - 674445714976086 T^{2} + \)\(20\!\cdots\!55\)\( T^{4} - \)\(38\!\cdots\!80\)\( T^{6} + \)\(20\!\cdots\!55\)\( p^{16} T^{8} - 674445714976086 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
61 \( ( 1 + 22989564 T + 678369782953575 T^{2} + \)\(87\!\cdots\!40\)\( T^{3} + 678369782953575 p^{8} T^{4} + 22989564 p^{16} T^{5} + p^{24} T^{6} )^{4} \)
67 \( ( 1 + 17763466977924 p T^{2} + \)\(83\!\cdots\!19\)\( T^{4} + \)\(42\!\cdots\!04\)\( T^{6} + \)\(83\!\cdots\!19\)\( p^{16} T^{8} + 17763466977924 p^{33} T^{10} + p^{48} T^{12} )^{2} \)
71 \( ( 1 - 249728727456006 T^{2} - \)\(17\!\cdots\!25\)\( T^{4} + \)\(18\!\cdots\!40\)\( T^{6} - \)\(17\!\cdots\!25\)\( p^{16} T^{8} - 249728727456006 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
73 \( ( 1 + 3594220472384358 T^{2} + \)\(57\!\cdots\!39\)\( T^{4} + \)\(77\!\cdots\!88\)\( p T^{6} + \)\(57\!\cdots\!39\)\( p^{16} T^{8} + 3594220472384358 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
79 \( ( 1 + 105100668 T + 8058279708186759 T^{2} + \)\(35\!\cdots\!64\)\( T^{3} + 8058279708186759 p^{8} T^{4} + 105100668 p^{16} T^{5} + p^{24} T^{6} )^{4} \)
83 \( ( 1 - 4613498548279092 T^{2} + \)\(20\!\cdots\!99\)\( T^{4} - \)\(46\!\cdots\!56\)\( T^{6} + \)\(20\!\cdots\!99\)\( p^{16} T^{8} - 4613498548279092 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
89 \( ( 1 - 15865425149725446 T^{2} + \)\(12\!\cdots\!55\)\( T^{4} - \)\(60\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!55\)\( p^{16} T^{8} - 15865425149725446 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
97 \( ( 1 + 17969647941417318 T^{2} + \)\(18\!\cdots\!79\)\( T^{4} + \)\(14\!\cdots\!44\)\( T^{6} + \)\(18\!\cdots\!79\)\( p^{16} T^{8} + 17969647941417318 p^{32} T^{10} + p^{48} T^{12} )^{2} \)
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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

−3.55977560602447279580805978957, −3.44016898265220291647694700570, −3.24681610238611521548134466646, −3.05570181712684594703639343264, −2.74343111127068317174360492061, −2.72406254266658986221811319024, −2.67786152408593906868975791471, −2.64625251761731333020611247777, −2.61462473586928723760611189116, −2.45988150680829466529372629106, −2.11457288427308884943804280718, −1.90000917744381977487161592445, −1.61941674416504217048393577095, −1.59188336925966382533154408814, −1.56723587389365935976091706167, −1.53954285011154934152506442039, −1.50219328447630645453638818810, −1.44624323598003177835797704541, −1.23477405910731412925185010396, −1.04530004070351562404125477837, −0.65135242644409299112134603565, −0.38917565127321539413348659027, −0.38429903719803325911130696861, −0.13582335946072630072296420768, −0.097994570561107832769375525520, 0.097994570561107832769375525520, 0.13582335946072630072296420768, 0.38429903719803325911130696861, 0.38917565127321539413348659027, 0.65135242644409299112134603565, 1.04530004070351562404125477837, 1.23477405910731412925185010396, 1.44624323598003177835797704541, 1.50219328447630645453638818810, 1.53954285011154934152506442039, 1.56723587389365935976091706167, 1.59188336925966382533154408814, 1.61941674416504217048393577095, 1.90000917744381977487161592445, 2.11457288427308884943804280718, 2.45988150680829466529372629106, 2.61462473586928723760611189116, 2.64625251761731333020611247777, 2.67786152408593906868975791471, 2.72406254266658986221811319024, 2.74343111127068317174360492061, 3.05570181712684594703639343264, 3.24681610238611521548134466646, 3.44016898265220291647694700570, 3.55977560602447279580805978957

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

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