Properties

Label 12-57e6-1.1-c1e6-0-2
Degree $12$
Conductor $34296447249$
Sign $1$
Analytic cond. $0.00889020$
Root an. cond. $0.674646$
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  + 3·2-s + 3·4-s − 6·5-s + 3·7-s − 8-s − 18·10-s + 3·11-s − 6·13-s + 9·14-s − 12·16-s − 12·17-s + 18·19-s − 18·20-s + 9·22-s − 9·23-s + 18·25-s − 18·26-s + 27-s + 9·28-s + 12·29-s − 24·31-s − 24·32-s − 36·34-s − 18·35-s + 12·37-s + 54·38-s + 6·40-s + ⋯
L(s)  = 1  + 2.12·2-s + 3/2·4-s − 2.68·5-s + 1.13·7-s − 0.353·8-s − 5.69·10-s + 0.904·11-s − 1.66·13-s + 2.40·14-s − 3·16-s − 2.91·17-s + 4.12·19-s − 4.02·20-s + 1.91·22-s − 1.87·23-s + 18/5·25-s − 3.53·26-s + 0.192·27-s + 1.70·28-s + 2.22·29-s − 4.31·31-s − 4.24·32-s − 6.17·34-s − 3.04·35-s + 1.97·37-s + 8.75·38-s + 0.948·40-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(0.00889020\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6965327444\)
\(L(\frac12)\) \(\approx\) \(0.6965327444\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{3} + T^{6} \)
19 \( 1 - 18 T + 162 T^{2} - 883 T^{3} + 162 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 - 3 T + 3 p T^{2} - p^{3} T^{3} + 15 T^{4} - 27 T^{5} + 47 T^{6} - 27 p T^{7} + 15 p^{2} T^{8} - p^{6} T^{9} + 3 p^{5} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 6 T + 18 T^{2} + 9 p T^{3} + 81 T^{4} + 87 T^{5} + 109 T^{6} + 87 p T^{7} + 81 p^{2} T^{8} + 9 p^{4} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
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

−9.015600755718425727796735086501, −8.379149423129916485373255258390, −8.249402599013603649521214469168, −7.966260175545973169527337362354, −7.81807022465561222102096894148, −7.76273569442966573181052283566, −7.45943078410886197661341543102, −7.10200700693017552283587643322, −7.08700500627690786029183846298, −6.64206074559130170624967366671, −6.57763013030256104277547696419, −6.03447793611712914005906134233, −5.94540037199828716197155307548, −5.44549826801597064547053299234, −5.02236294594966168153223302238, −4.85626218196179768684971916420, −4.83644641260977243684267893874, −4.46851647850660291802023112003, −4.34511705558270003881714561954, −4.11590931235712046424992216906, −3.63485472156776633607676744889, −3.39060129821530087042324845572, −3.22988379104188754171975527228, −2.42377951463254828862478205074, −2.12470280408781661591321108240, 2.12470280408781661591321108240, 2.42377951463254828862478205074, 3.22988379104188754171975527228, 3.39060129821530087042324845572, 3.63485472156776633607676744889, 4.11590931235712046424992216906, 4.34511705558270003881714561954, 4.46851647850660291802023112003, 4.83644641260977243684267893874, 4.85626218196179768684971916420, 5.02236294594966168153223302238, 5.44549826801597064547053299234, 5.94540037199828716197155307548, 6.03447793611712914005906134233, 6.57763013030256104277547696419, 6.64206074559130170624967366671, 7.08700500627690786029183846298, 7.10200700693017552283587643322, 7.45943078410886197661341543102, 7.76273569442966573181052283566, 7.81807022465561222102096894148, 7.966260175545973169527337362354, 8.249402599013603649521214469168, 8.379149423129916485373255258390, 9.015600755718425727796735086501

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

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