Properties

Degree $24$
Conductor $5.960\times 10^{16}$
Sign $1$
Motivic weight $33$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.20e10·4-s + 1.97e16·9-s + 3.01e17·11-s + 1.47e20·16-s − 3.42e21·19-s − 2.61e24·29-s + 2.16e25·31-s + 4.34e26·36-s + 8.96e26·41-s + 6.63e27·44-s + 5.35e28·49-s + 3.56e29·59-s − 5.64e29·61-s − 7.63e29·64-s + 1.69e31·71-s − 7.53e31·76-s + 4.41e31·79-s + 9.06e31·81-s + 6.72e32·89-s + 5.95e33·99-s + 1.96e33·101-s + 1.33e34·109-s − 5.74e34·116-s − 8.33e34·121-s + 4.76e35·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 2.56·4-s + 3.55·9-s + 1.97·11-s + 2.00·16-s − 2.72·19-s − 1.93·29-s + 5.34·31-s + 9.10·36-s + 2.19·41-s + 5.07·44-s + 6.92·49-s + 2.15·59-s − 1.96·61-s − 1.20·64-s + 4.81·71-s − 6.97·76-s + 2.15·79-s + 2.93·81-s + 4.60·89-s + 7.02·99-s + 1.66·101-s + 3.20·109-s − 4.96·116-s − 3.58·121-s + 13.7·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(34-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24}\right)^{s/2} \, \Gamma_{\C}(s+33/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(5^{24}\)
Sign: $1$
Motivic weight: \(33\)
Character: induced by $\chi_{25} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 5^{24} ,\ ( \ : [33/2]^{12} ),\ 1 )\)

Particular Values

\(L(17)\) \(\approx\) \(163.0489029\)
\(L(\frac12)\) \(\approx\) \(163.0489029\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 - 5504740475 p^{2} T^{2} + 5264871706344391681 p^{6} T^{4} - \)\(20\!\cdots\!75\)\( p^{14} T^{6} + \)\(90\!\cdots\!35\)\( p^{38} T^{8} - \)\(24\!\cdots\!25\)\( p^{49} T^{10} + \)\(17\!\cdots\!45\)\( p^{62} T^{12} - \)\(24\!\cdots\!25\)\( p^{115} T^{14} + \)\(90\!\cdots\!35\)\( p^{170} T^{16} - \)\(20\!\cdots\!75\)\( p^{212} T^{18} + 5264871706344391681 p^{270} T^{20} - 5504740475 p^{332} T^{22} + p^{396} T^{24} \)
3 \( 1 - 19752260815987700 T^{2} + \)\(33\!\cdots\!86\)\( p^{2} T^{4} - \)\(47\!\cdots\!00\)\( p^{8} T^{6} + \)\(70\!\cdots\!35\)\( p^{18} T^{8} - \)\(94\!\cdots\!00\)\( p^{30} T^{10} + \)\(12\!\cdots\!80\)\( p^{44} T^{12} - \)\(94\!\cdots\!00\)\( p^{96} T^{14} + \)\(70\!\cdots\!35\)\( p^{150} T^{16} - \)\(47\!\cdots\!00\)\( p^{206} T^{18} + \)\(33\!\cdots\!86\)\( p^{266} T^{20} - 19752260815987700 p^{330} T^{22} + p^{396} T^{24} \)
7 \( 1 - \)\(53\!\cdots\!00\)\( T^{2} + \)\(29\!\cdots\!06\)\( p^{2} T^{4} - \)\(10\!\cdots\!00\)\( p^{4} T^{6} + \)\(28\!\cdots\!35\)\( p^{6} T^{8} - \)\(25\!\cdots\!00\)\( p^{12} T^{10} + \)\(37\!\cdots\!80\)\( p^{20} T^{12} - \)\(25\!\cdots\!00\)\( p^{78} T^{14} + \)\(28\!\cdots\!35\)\( p^{138} T^{16} - \)\(10\!\cdots\!00\)\( p^{202} T^{18} + \)\(29\!\cdots\!06\)\( p^{266} T^{20} - \)\(53\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
11 \( ( 1 - 113268892979632 p^{3} T + \)\(62\!\cdots\!26\)\( p^{2} T^{2} - \)\(36\!\cdots\!20\)\( p^{3} T^{3} + \)\(13\!\cdots\!45\)\( p^{5} T^{4} - \)\(59\!\cdots\!92\)\( p^{5} T^{5} + \)\(25\!\cdots\!44\)\( p^{6} T^{6} - \)\(59\!\cdots\!92\)\( p^{38} T^{7} + \)\(13\!\cdots\!45\)\( p^{71} T^{8} - \)\(36\!\cdots\!20\)\( p^{102} T^{9} + \)\(62\!\cdots\!26\)\( p^{134} T^{10} - 113268892979632 p^{168} T^{11} + p^{198} T^{12} )^{2} \)
13 \( 1 - \)\(43\!\cdots\!00\)\( T^{2} + \)\(95\!\cdots\!54\)\( T^{4} - \)\(82\!\cdots\!00\)\( p^{2} T^{6} + \)\(31\!\cdots\!35\)\( p^{6} T^{8} - \)\(89\!\cdots\!00\)\( p^{10} T^{10} + \)\(12\!\cdots\!80\)\( p^{16} T^{12} - \)\(89\!\cdots\!00\)\( p^{76} T^{14} + \)\(31\!\cdots\!35\)\( p^{138} T^{16} - \)\(82\!\cdots\!00\)\( p^{200} T^{18} + \)\(95\!\cdots\!54\)\( p^{264} T^{20} - \)\(43\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
17 \( 1 - \)\(12\!\cdots\!00\)\( T^{2} + \)\(94\!\cdots\!14\)\( T^{4} - \)\(17\!\cdots\!00\)\( p^{2} T^{6} + \)\(29\!\cdots\!15\)\( p^{4} T^{8} - \)\(47\!\cdots\!00\)\( p^{6} T^{10} + \)\(68\!\cdots\!80\)\( p^{8} T^{12} - \)\(47\!\cdots\!00\)\( p^{72} T^{14} + \)\(29\!\cdots\!15\)\( p^{136} T^{16} - \)\(17\!\cdots\!00\)\( p^{200} T^{18} + \)\(94\!\cdots\!14\)\( p^{264} T^{20} - \)\(12\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
19 \( ( 1 + \)\(17\!\cdots\!00\)\( T + \)\(41\!\cdots\!54\)\( T^{2} + \)\(30\!\cdots\!00\)\( p T^{3} + \)\(33\!\cdots\!15\)\( p^{2} T^{4} + \)\(21\!\cdots\!00\)\( p^{3} T^{5} + \)\(17\!\cdots\!80\)\( p^{4} T^{6} + \)\(21\!\cdots\!00\)\( p^{36} T^{7} + \)\(33\!\cdots\!15\)\( p^{68} T^{8} + \)\(30\!\cdots\!00\)\( p^{100} T^{9} + \)\(41\!\cdots\!54\)\( p^{132} T^{10} + \)\(17\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} )^{2} \)
23 \( 1 - \)\(44\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!34\)\( T^{4} - \)\(43\!\cdots\!00\)\( p^{2} T^{6} + \)\(11\!\cdots\!15\)\( p^{4} T^{8} - \)\(26\!\cdots\!00\)\( p^{6} T^{10} + \)\(48\!\cdots\!80\)\( p^{8} T^{12} - \)\(26\!\cdots\!00\)\( p^{72} T^{14} + \)\(11\!\cdots\!15\)\( p^{136} T^{16} - \)\(43\!\cdots\!00\)\( p^{200} T^{18} + \)\(12\!\cdots\!34\)\( p^{264} T^{20} - \)\(44\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
29 \( ( 1 + \)\(13\!\cdots\!00\)\( T + \)\(12\!\cdots\!46\)\( p T^{2} + \)\(73\!\cdots\!00\)\( p^{2} T^{3} + \)\(39\!\cdots\!35\)\( p^{3} T^{4} + \)\(25\!\cdots\!00\)\( p^{4} T^{5} + \)\(10\!\cdots\!20\)\( p^{5} T^{6} + \)\(25\!\cdots\!00\)\( p^{37} T^{7} + \)\(39\!\cdots\!35\)\( p^{69} T^{8} + \)\(73\!\cdots\!00\)\( p^{101} T^{9} + \)\(12\!\cdots\!46\)\( p^{133} T^{10} + \)\(13\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} )^{2} \)
31 \( ( 1 - \)\(34\!\cdots\!92\)\( p T + \)\(44\!\cdots\!66\)\( p^{3} T^{2} - \)\(30\!\cdots\!20\)\( p^{3} T^{3} + \)\(66\!\cdots\!95\)\( p^{4} T^{4} - \)\(10\!\cdots\!92\)\( p^{5} T^{5} + \)\(15\!\cdots\!84\)\( p^{6} T^{6} - \)\(10\!\cdots\!92\)\( p^{38} T^{7} + \)\(66\!\cdots\!95\)\( p^{70} T^{8} - \)\(30\!\cdots\!20\)\( p^{102} T^{9} + \)\(44\!\cdots\!66\)\( p^{135} T^{10} - \)\(34\!\cdots\!92\)\( p^{166} T^{11} + p^{198} T^{12} )^{2} \)
37 \( 1 - \)\(33\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!54\)\( T^{4} - \)\(43\!\cdots\!00\)\( T^{6} + \)\(66\!\cdots\!15\)\( T^{8} - \)\(23\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!80\)\( T^{12} - \)\(23\!\cdots\!00\)\( p^{66} T^{14} + \)\(66\!\cdots\!15\)\( p^{132} T^{16} - \)\(43\!\cdots\!00\)\( p^{198} T^{18} + \)\(11\!\cdots\!54\)\( p^{264} T^{20} - \)\(33\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
41 \( ( 1 - \)\(44\!\cdots\!32\)\( T + \)\(40\!\cdots\!86\)\( T^{2} - \)\(20\!\cdots\!20\)\( T^{3} + \)\(95\!\cdots\!95\)\( T^{4} - \)\(34\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!64\)\( T^{6} - \)\(34\!\cdots\!92\)\( p^{33} T^{7} + \)\(95\!\cdots\!95\)\( p^{66} T^{8} - \)\(20\!\cdots\!20\)\( p^{99} T^{9} + \)\(40\!\cdots\!86\)\( p^{132} T^{10} - \)\(44\!\cdots\!32\)\( p^{165} T^{11} + p^{198} T^{12} )^{2} \)
43 \( 1 - \)\(71\!\cdots\!00\)\( T^{2} + \)\(24\!\cdots\!94\)\( T^{4} - \)\(52\!\cdots\!00\)\( T^{6} + \)\(81\!\cdots\!15\)\( T^{8} - \)\(95\!\cdots\!00\)\( T^{10} + \)\(86\!\cdots\!80\)\( T^{12} - \)\(95\!\cdots\!00\)\( p^{66} T^{14} + \)\(81\!\cdots\!15\)\( p^{132} T^{16} - \)\(52\!\cdots\!00\)\( p^{198} T^{18} + \)\(24\!\cdots\!94\)\( p^{264} T^{20} - \)\(71\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
47 \( 1 - \)\(34\!\cdots\!00\)\( T^{2} + \)\(31\!\cdots\!74\)\( T^{4} + \)\(71\!\cdots\!00\)\( T^{6} - \)\(13\!\cdots\!85\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{10} + \)\(68\!\cdots\!80\)\( T^{12} - \)\(18\!\cdots\!00\)\( p^{66} T^{14} - \)\(13\!\cdots\!85\)\( p^{132} T^{16} + \)\(71\!\cdots\!00\)\( p^{198} T^{18} + \)\(31\!\cdots\!74\)\( p^{264} T^{20} - \)\(34\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
53 \( 1 - \)\(24\!\cdots\!00\)\( T^{2} + \)\(41\!\cdots\!74\)\( T^{4} - \)\(49\!\cdots\!00\)\( T^{6} + \)\(49\!\cdots\!15\)\( T^{8} - \)\(41\!\cdots\!00\)\( T^{10} + \)\(34\!\cdots\!80\)\( T^{12} - \)\(41\!\cdots\!00\)\( p^{66} T^{14} + \)\(49\!\cdots\!15\)\( p^{132} T^{16} - \)\(49\!\cdots\!00\)\( p^{198} T^{18} + \)\(41\!\cdots\!74\)\( p^{264} T^{20} - \)\(24\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
59 \( ( 1 - \)\(17\!\cdots\!00\)\( T + \)\(85\!\cdots\!74\)\( T^{2} - \)\(90\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!15\)\( T^{4} - \)\(91\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!80\)\( T^{6} - \)\(91\!\cdots\!00\)\( p^{33} T^{7} + \)\(19\!\cdots\!15\)\( p^{66} T^{8} - \)\(90\!\cdots\!00\)\( p^{99} T^{9} + \)\(85\!\cdots\!74\)\( p^{132} T^{10} - \)\(17\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} )^{2} \)
61 \( ( 1 + \)\(28\!\cdots\!08\)\( T + \)\(42\!\cdots\!46\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(80\!\cdots\!95\)\( T^{4} + \)\(16\!\cdots\!08\)\( T^{5} + \)\(85\!\cdots\!84\)\( T^{6} + \)\(16\!\cdots\!08\)\( p^{33} T^{7} + \)\(80\!\cdots\!95\)\( p^{66} T^{8} + \)\(10\!\cdots\!80\)\( p^{99} T^{9} + \)\(42\!\cdots\!46\)\( p^{132} T^{10} + \)\(28\!\cdots\!08\)\( p^{165} T^{11} + p^{198} T^{12} )^{2} \)
67 \( 1 - \)\(68\!\cdots\!00\)\( T^{2} + \)\(26\!\cdots\!14\)\( T^{4} - \)\(83\!\cdots\!00\)\( T^{6} + \)\(22\!\cdots\!15\)\( T^{8} - \)\(51\!\cdots\!00\)\( T^{10} + \)\(98\!\cdots\!80\)\( T^{12} - \)\(51\!\cdots\!00\)\( p^{66} T^{14} + \)\(22\!\cdots\!15\)\( p^{132} T^{16} - \)\(83\!\cdots\!00\)\( p^{198} T^{18} + \)\(26\!\cdots\!14\)\( p^{264} T^{20} - \)\(68\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
71 \( ( 1 - \)\(84\!\cdots\!72\)\( T + \)\(11\!\cdots\!06\)\( p T^{2} - \)\(44\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!95\)\( T^{4} - \)\(99\!\cdots\!92\)\( T^{5} + \)\(39\!\cdots\!44\)\( T^{6} - \)\(99\!\cdots\!92\)\( p^{33} T^{7} + \)\(24\!\cdots\!95\)\( p^{66} T^{8} - \)\(44\!\cdots\!20\)\( p^{99} T^{9} + \)\(11\!\cdots\!06\)\( p^{133} T^{10} - \)\(84\!\cdots\!72\)\( p^{165} T^{11} + p^{198} T^{12} )^{2} \)
73 \( 1 - \)\(24\!\cdots\!00\)\( T^{2} + \)\(29\!\cdots\!34\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!15\)\( T^{8} - \)\(61\!\cdots\!00\)\( T^{10} + \)\(21\!\cdots\!80\)\( T^{12} - \)\(61\!\cdots\!00\)\( p^{66} T^{14} + \)\(13\!\cdots\!15\)\( p^{132} T^{16} - \)\(23\!\cdots\!00\)\( p^{198} T^{18} + \)\(29\!\cdots\!34\)\( p^{264} T^{20} - \)\(24\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
79 \( ( 1 - \)\(22\!\cdots\!00\)\( T + \)\(23\!\cdots\!34\)\( T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!15\)\( T^{4} - \)\(34\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} - \)\(34\!\cdots\!00\)\( p^{33} T^{7} + \)\(23\!\cdots\!15\)\( p^{66} T^{8} - \)\(43\!\cdots\!00\)\( p^{99} T^{9} + \)\(23\!\cdots\!34\)\( p^{132} T^{10} - \)\(22\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} )^{2} \)
83 \( 1 - \)\(15\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!14\)\( T^{4} - \)\(62\!\cdots\!00\)\( T^{6} + \)\(24\!\cdots\!15\)\( T^{8} - \)\(71\!\cdots\!00\)\( T^{10} + \)\(17\!\cdots\!80\)\( T^{12} - \)\(71\!\cdots\!00\)\( p^{66} T^{14} + \)\(24\!\cdots\!15\)\( p^{132} T^{16} - \)\(62\!\cdots\!00\)\( p^{198} T^{18} + \)\(12\!\cdots\!14\)\( p^{264} T^{20} - \)\(15\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
89 \( ( 1 - \)\(33\!\cdots\!00\)\( T + \)\(10\!\cdots\!14\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!15\)\( T^{4} - \)\(68\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} - \)\(68\!\cdots\!00\)\( p^{33} T^{7} + \)\(40\!\cdots\!15\)\( p^{66} T^{8} - \)\(20\!\cdots\!00\)\( p^{99} T^{9} + \)\(10\!\cdots\!14\)\( p^{132} T^{10} - \)\(33\!\cdots\!00\)\( p^{165} T^{11} + p^{198} T^{12} )^{2} \)
97 \( 1 - \)\(19\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!74\)\( T^{4} - \)\(86\!\cdots\!00\)\( T^{6} + \)\(38\!\cdots\!15\)\( T^{8} - \)\(17\!\cdots\!00\)\( T^{10} + \)\(73\!\cdots\!80\)\( T^{12} - \)\(17\!\cdots\!00\)\( p^{66} T^{14} + \)\(38\!\cdots\!15\)\( p^{132} T^{16} - \)\(86\!\cdots\!00\)\( p^{198} T^{18} + \)\(16\!\cdots\!74\)\( p^{264} T^{20} - \)\(19\!\cdots\!00\)\( p^{330} T^{22} + p^{396} T^{24} \)
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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

−2.40329039385924943791021453324, −2.23514204748033604339471587156, −2.23095275252770717189151493754, −2.11302401329779819948821669578, −2.06667259525978232056771891243, −1.91358164485269753717791077145, −1.89562006490296423106431805270, −1.84204087753400759778753592811, −1.83697122514945304294950225978, −1.72488636450738647619483916293, −1.54662376340999003432521017151, −1.49258555100176961134242711108, −1.22450504257602700691331416713, −1.19812121222543098061826626943, −0.925547184565010903530651526682, −0.922407971207842133292003437468, −0.904082207925808231484590523876, −0.71428932532107507233888450870, −0.69264242345870491518276435380, −0.62082918756665698136625786067, −0.61292131097700950164597956546, −0.52444599838672861662399858791, −0.50016107173652805996046967018, −0.45357142848253631996093133588, −0.03868770458918721846629333644, 0.03868770458918721846629333644, 0.45357142848253631996093133588, 0.50016107173652805996046967018, 0.52444599838672861662399858791, 0.61292131097700950164597956546, 0.62082918756665698136625786067, 0.69264242345870491518276435380, 0.71428932532107507233888450870, 0.904082207925808231484590523876, 0.922407971207842133292003437468, 0.925547184565010903530651526682, 1.19812121222543098061826626943, 1.22450504257602700691331416713, 1.49258555100176961134242711108, 1.54662376340999003432521017151, 1.72488636450738647619483916293, 1.83697122514945304294950225978, 1.84204087753400759778753592811, 1.89562006490296423106431805270, 1.91358164485269753717791077145, 2.06667259525978232056771891243, 2.11302401329779819948821669578, 2.23095275252770717189151493754, 2.23514204748033604339471587156, 2.40329039385924943791021453324

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

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