Properties

Label 12-912e6-1.1-c1e6-0-5
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $149152.$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 3·7-s − 3·11-s − 6·13-s − 12·17-s − 18·19-s + 9·23-s + 18·25-s − 27-s + 12·29-s + 24·31-s + 18·35-s + 12·37-s + 6·41-s − 18·43-s + 33·47-s + 21·49-s − 24·53-s + 18·55-s − 21·61-s + 36·65-s − 30·67-s − 18·71-s − 27·73-s + 9·77-s − 12·79-s − 15·83-s + ⋯
L(s)  = 1  − 2.68·5-s − 1.13·7-s − 0.904·11-s − 1.66·13-s − 2.91·17-s − 4.12·19-s + 1.87·23-s + 18/5·25-s − 0.192·27-s + 2.22·29-s + 4.31·31-s + 3.04·35-s + 1.97·37-s + 0.937·41-s − 2.74·43-s + 4.81·47-s + 3·49-s − 3.29·53-s + 2.42·55-s − 2.68·61-s + 4.46·65-s − 3.66·67-s − 2.13·71-s − 3.16·73-s + 1.02·77-s − 1.35·79-s − 1.64·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(149152.\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3924803854\)
\(L(\frac12)\) \(\approx\) \(0.3924803854\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T^{3} + T^{6} \)
19 \( 1 + 18 T + 162 T^{2} + 883 T^{3} + 162 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + 6 T + 18 T^{2} + 9 p T^{3} + 81 T^{4} + 87 T^{5} + 109 T^{6} + 87 p T^{7} + 81 p^{2} T^{8} + 9 p^{4} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 3 T - 12 T^{2} - 19 T^{3} + 171 T^{4} + 18 p T^{5} - 1161 T^{6} + 18 p^{2} T^{7} + 171 p^{2} T^{8} - 19 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 T - 6 T^{2} - 13 T^{3} - 27 T^{4} - 192 T^{5} - 61 T^{6} - 192 p T^{7} - 27 p^{2} T^{8} - 13 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 6 T + 30 T^{2} + 86 T^{3} + 288 T^{4} + 180 T^{5} + 231 T^{6} + 180 p T^{7} + 288 p^{2} T^{8} + 86 p^{3} T^{9} + 30 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 12 T + 114 T^{2} + 809 T^{3} + 4965 T^{4} + 25191 T^{5} + 111881 T^{6} + 25191 p T^{7} + 4965 p^{2} T^{8} + 809 p^{3} T^{9} + 114 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 9 T + 18 T^{2} + 130 T^{3} - 603 T^{4} - 4725 T^{5} + 49121 T^{6} - 4725 p T^{7} - 603 p^{2} T^{8} + 130 p^{3} T^{9} + 18 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 12 T + 60 T^{2} - 26 T^{3} - 1200 T^{4} + 9792 T^{5} - 51073 T^{6} + 9792 p T^{7} - 1200 p^{2} T^{8} - 26 p^{3} T^{9} + 60 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 24 T + 294 T^{2} - 2814 T^{3} + 23334 T^{4} - 160962 T^{5} + 949331 T^{6} - 160962 p T^{7} + 23334 p^{2} T^{8} - 2814 p^{3} T^{9} + 294 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 6 T + 102 T^{2} - 393 T^{3} + 102 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 6 T + 81 T^{2} - 351 T^{3} + 4257 T^{4} - 12003 T^{5} + 138106 T^{6} - 12003 p T^{7} + 4257 p^{2} T^{8} - 351 p^{3} T^{9} + 81 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 18 T + 144 T^{2} + 520 T^{3} + 2610 T^{4} + 49554 T^{5} + 474393 T^{6} + 49554 p T^{7} + 2610 p^{2} T^{8} + 520 p^{3} T^{9} + 144 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 33 T + 477 T^{2} - 3357 T^{3} + 648 T^{4} + 230538 T^{5} - 2370995 T^{6} + 230538 p T^{7} + 648 p^{2} T^{8} - 3357 p^{3} T^{9} + 477 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 24 T + 375 T^{2} + 4609 T^{3} + 47013 T^{4} + 413955 T^{5} + 3190646 T^{6} + 413955 p T^{7} + 47013 p^{2} T^{8} + 4609 p^{3} T^{9} + 375 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 54 T^{2} - 661 T^{3} + 5391 T^{4} - 27351 T^{5} + 485585 T^{6} - 27351 p T^{7} + 5391 p^{2} T^{8} - 661 p^{3} T^{9} + 54 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 + 21 T + 204 T^{2} + 872 T^{3} - 5166 T^{4} - 119673 T^{5} - 1122129 T^{6} - 119673 p T^{7} - 5166 p^{2} T^{8} + 872 p^{3} T^{9} + 204 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 30 T + 372 T^{2} + 1966 T^{3} - 7344 T^{4} - 261360 T^{5} - 2850267 T^{6} - 261360 p T^{7} - 7344 p^{2} T^{8} + 1966 p^{3} T^{9} + 372 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 18 T + 144 T^{2} + 180 T^{3} - 4734 T^{4} - 88470 T^{5} - 842507 T^{6} - 88470 p T^{7} - 4734 p^{2} T^{8} + 180 p^{3} T^{9} + 144 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 27 T + 324 T^{2} + 2356 T^{3} + 4887 T^{4} - 141291 T^{5} - 1960287 T^{6} - 141291 p T^{7} + 4887 p^{2} T^{8} + 2356 p^{3} T^{9} + 324 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 12 T + 168 T^{2} + 1772 T^{3} + 24948 T^{4} + 218700 T^{5} + 2149713 T^{6} + 218700 p T^{7} + 24948 p^{2} T^{8} + 1772 p^{3} T^{9} + 168 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 15 T - 60 T^{2} - 603 T^{3} + 20985 T^{4} + 88800 T^{5} - 1199405 T^{6} + 88800 p T^{7} + 20985 p^{2} T^{8} - 603 p^{3} T^{9} - 60 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 45 T + 900 T^{2} - 9000 T^{3} + 10575 T^{4} + 1004445 T^{5} - 14638319 T^{6} + 1004445 p T^{7} + 10575 p^{2} T^{8} - 9000 p^{3} T^{9} + 900 p^{4} T^{10} - 45 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 21 T + 210 T^{2} - 814 T^{3} - 6003 T^{4} + 250281 T^{5} - 3257067 T^{6} + 250281 p T^{7} - 6003 p^{2} T^{8} - 814 p^{3} T^{9} + 210 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.43386871529291280421038336886, −4.82591730706654812319543906288, −4.68841416822256704722450716004, −4.66506504410911652670285397487, −4.66042631994961041643595079180, −4.46803042865202192300336250430, −4.37312605925283106039698516381, −4.34762492947556215581268675971, −4.29468420483364127593914819270, −4.04384730604352239424660490783, −3.62190019770505567624735843292, −3.47686801783567870793601811438, −3.21970403988830157734174590187, −2.98947880895745698162157459260, −2.72726558111037349015559956817, −2.63283738872861810648007092098, −2.56429655864707699771904598626, −2.54145618089958858166496002491, −2.47725029048930036292738339545, −1.77508735537295722399337411194, −1.55276527064776908192171223178, −1.27276598099551737536911257629, −0.62178652312681625238999218905, −0.40793135329163387231086777620, −0.28009772849151320320362714088, 0.28009772849151320320362714088, 0.40793135329163387231086777620, 0.62178652312681625238999218905, 1.27276598099551737536911257629, 1.55276527064776908192171223178, 1.77508735537295722399337411194, 2.47725029048930036292738339545, 2.54145618089958858166496002491, 2.56429655864707699771904598626, 2.63283738872861810648007092098, 2.72726558111037349015559956817, 2.98947880895745698162157459260, 3.21970403988830157734174590187, 3.47686801783567870793601811438, 3.62190019770505567624735843292, 4.04384730604352239424660490783, 4.29468420483364127593914819270, 4.34762492947556215581268675971, 4.37312605925283106039698516381, 4.46803042865202192300336250430, 4.66042631994961041643595079180, 4.66506504410911652670285397487, 4.68841416822256704722450716004, 4.82591730706654812319543906288, 5.43386871529291280421038336886

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

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