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Properties

Label 12-99e6-1.1-c1e6-0-0
Degree $12$
Conductor $941480149401$
Sign $1$
Analytic cond. $0.244047$
Root an. cond. $0.889111$
Motivic weight $1$
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  + 3·4-s − 3·5-s + 3·7-s − 2·8-s + 3·11-s + 9·13-s + 6·16-s − 6·17-s − 18·19-s − 9·20-s + 6·23-s + 15·25-s − 9·27-s + 9·28-s + 6·29-s + 9·31-s − 9·32-s − 9·35-s − 24·37-s + 6·40-s + 6·41-s − 3·43-s + 9·44-s − 12·47-s + 21·49-s + 27·52-s + 6·53-s + ⋯
L(s)  = 1  + 3/2·4-s − 1.34·5-s + 1.13·7-s − 0.707·8-s + 0.904·11-s + 2.49·13-s + 3/2·16-s − 1.45·17-s − 4.12·19-s − 2.01·20-s + 1.25·23-s + 3·25-s − 1.73·27-s + 1.70·28-s + 1.11·29-s + 1.61·31-s − 1.59·32-s − 1.52·35-s − 3.94·37-s + 0.948·40-s + 0.937·41-s − 0.457·43-s + 1.35·44-s − 1.75·47-s + 3·49-s + 3.74·52-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(0.244047\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.152877728\)
\(L(\frac12)\) \(\approx\) \(1.152877728\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p^{2} T^{3} + p^{3} T^{6} \)
11 \( ( 1 - T + T^{2} )^{3} \)
good2 \( 1 - 3 T^{2} + p T^{3} + 3 T^{4} - 3 T^{5} - T^{6} - 3 p T^{7} + 3 p^{2} T^{8} + p^{4} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 + 3 T - 6 T^{2} - 13 T^{3} + 63 T^{4} + 12 p T^{5} - 259 T^{6} + 12 p^{2} T^{7} + 63 p^{2} T^{8} - 13 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 3 T - 12 T^{2} + 19 T^{3} + 171 T^{4} - 18 p T^{5} - 1161 T^{6} - 18 p^{2} T^{7} + 171 p^{2} T^{8} + 19 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 9 T + 27 T^{2} - 20 T^{3} + 9 p T^{4} - 1467 T^{5} + 7086 T^{6} - 1467 p T^{7} + 9 p^{3} T^{8} - 20 p^{3} T^{9} + 27 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 + 3 T + 45 T^{2} + 103 T^{3} + 45 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 9 T + 75 T^{2} + 333 T^{3} + 75 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 6 T - 36 T^{2} + 82 T^{3} + 1986 T^{4} - 1806 T^{5} - 47893 T^{6} - 1806 p T^{7} + 1986 p^{2} T^{8} + 82 p^{3} T^{9} - 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 6 T - 6 T^{2} + 18 T^{3} - 264 T^{4} + 3900 T^{5} - 14429 T^{6} + 3900 p T^{7} - 264 p^{2} T^{8} + 18 p^{3} T^{9} - 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 9 T - 18 T^{2} + 187 T^{3} + 1881 T^{4} - 6768 T^{5} - 31569 T^{6} - 6768 p T^{7} + 1881 p^{2} T^{8} + 187 p^{3} T^{9} - 18 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 12 T + 138 T^{2} + 885 T^{3} + 138 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 6 T - 6 T^{2} - 202 T^{3} - 108 T^{4} + 9528 T^{5} + 5627 T^{6} + 9528 p T^{7} - 108 p^{2} T^{8} - 202 p^{3} T^{9} - 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T + 24 T^{2} + 525 T^{3} + 1725 T^{4} + 13692 T^{5} + 213923 T^{6} + 13692 p T^{7} + 1725 p^{2} T^{8} + 525 p^{3} T^{9} + 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 12 T + 18 T^{2} - 386 T^{3} - 1434 T^{4} + 3954 T^{5} + 43979 T^{6} + 3954 p T^{7} - 1434 p^{2} T^{8} - 386 p^{3} T^{9} + 18 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 3 T + 33 T^{2} - 261 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 21 T + 198 T^{2} + 1081 T^{3} + 2481 T^{4} - 40380 T^{5} - 564133 T^{6} - 40380 p T^{7} + 2481 p^{2} T^{8} + 1081 p^{3} T^{9} + 198 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 21 T + 114 T^{2} - 1101 T^{3} + 26439 T^{4} - 183108 T^{5} + 602021 T^{6} - 183108 p T^{7} + 26439 p^{2} T^{8} - 1101 p^{3} T^{9} + 114 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 3 T - 66 T^{2} + 575 T^{3} + 1125 T^{4} - 29808 T^{5} + 214875 T^{6} - 29808 p T^{7} + 1125 p^{2} T^{8} + 575 p^{3} T^{9} - 66 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 12 T + 222 T^{2} - 1593 T^{3} + 222 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 6 T + 114 T^{2} + 173 T^{3} + 114 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 9 T - 180 T^{2} + 533 T^{3} + 34533 T^{4} - 61128 T^{5} - 2824521 T^{6} - 61128 p T^{7} + 34533 p^{2} T^{8} + 533 p^{3} T^{9} - 180 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 222 T^{2} - 54 T^{3} + 30858 T^{4} + 5994 T^{5} - 2946269 T^{6} + 5994 p T^{7} + 30858 p^{2} T^{8} - 54 p^{3} T^{9} - 222 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 12 T + 159 T^{2} + 1728 T^{3} + 159 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 3 T - 141 T^{2} - 724 T^{3} + 8361 T^{4} + 79623 T^{5} - 423066 T^{6} + 79623 p T^{7} + 8361 p^{2} T^{8} - 724 p^{3} T^{9} - 141 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−8.033488627298117695249270513392, −7.76827769790549747607730812702, −7.17809891426627090209318555779, −7.16576415121361910830222974071, −7.07013772828168894407102789621, −6.82326055833356684491144028129, −6.48801023790524987408416705268, −6.39750311496769766351948640263, −6.32538323149485518983050714305, −6.28886996829218224912112058240, −5.72053030813267742374670059799, −5.44036792417210398304367993872, −5.17843851421376892230602417199, −4.91702843983405519247896985251, −4.72213850669722601074317444081, −4.13069803706824618099025629107, −4.07484953111814333481348863115, −3.95841629004373223610675564355, −3.73666614660862836087454104850, −3.31033569234948605735922775028, −2.90466115875686840687765127827, −2.42509581480708884265128443841, −2.37101318364925647059534818824, −1.57470559883870459294167480629, −1.43255679275398639033860616590, 1.43255679275398639033860616590, 1.57470559883870459294167480629, 2.37101318364925647059534818824, 2.42509581480708884265128443841, 2.90466115875686840687765127827, 3.31033569234948605735922775028, 3.73666614660862836087454104850, 3.95841629004373223610675564355, 4.07484953111814333481348863115, 4.13069803706824618099025629107, 4.72213850669722601074317444081, 4.91702843983405519247896985251, 5.17843851421376892230602417199, 5.44036792417210398304367993872, 5.72053030813267742374670059799, 6.28886996829218224912112058240, 6.32538323149485518983050714305, 6.39750311496769766351948640263, 6.48801023790524987408416705268, 6.82326055833356684491144028129, 7.07013772828168894407102789621, 7.16576415121361910830222974071, 7.17809891426627090209318555779, 7.76827769790549747607730812702, 8.033488627298117695249270513392

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

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