Properties

Label 12-273e6-1.1-c1e6-0-2
Degree $12$
Conductor $4.140\times 10^{14}$
Sign $1$
Analytic cond. $107.309$
Root an. cond. $1.47645$
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  − 3·3-s + 3·4-s + 12·5-s + 3·7-s − 2·8-s + 3·9-s − 6·11-s − 9·12-s − 36·15-s + 6·16-s − 6·17-s − 9·19-s + 36·20-s − 9·21-s − 3·23-s + 6·24-s + 60·25-s + 2·27-s + 9·28-s − 9·29-s + 18·31-s − 9·32-s + 18·33-s + 36·35-s + 9·36-s − 12·37-s − 24·40-s + ⋯
L(s)  = 1  − 1.73·3-s + 3/2·4-s + 5.36·5-s + 1.13·7-s − 0.707·8-s + 9-s − 1.80·11-s − 2.59·12-s − 9.29·15-s + 3/2·16-s − 1.45·17-s − 2.06·19-s + 8.04·20-s − 1.96·21-s − 0.625·23-s + 1.22·24-s + 12·25-s + 0.384·27-s + 1.70·28-s − 1.67·29-s + 3.23·31-s − 1.59·32-s + 3.13·33-s + 6.08·35-s + 3/2·36-s − 1.97·37-s − 3.79·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 7^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(107.309\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 7^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.701564215\)
\(L(\frac12)\) \(\approx\) \(3.701564215\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T + T^{2} )^{3} \)
7 \( ( 1 - T + T^{2} )^{3} \)
13 \( 1 - 19 T^{3} + p^{3} T^{6} \)
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 \( ( 1 - 6 T + 24 T^{2} - 61 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( 1 + 6 T + 12 T^{2} + 2 p T^{3} - 54 T^{4} - 978 T^{5} - 4309 T^{6} - 978 p T^{7} - 54 p^{2} T^{8} + 2 p^{4} T^{9} + 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 6 T + 12 T^{2} + 54 T^{3} - 6 p T^{4} - 2082 T^{5} - 8345 T^{6} - 2082 p T^{7} - 6 p^{3} T^{8} + 54 p^{3} T^{9} + 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 9 T + 9 T^{2} - 2 T^{3} + 891 T^{4} + 2025 T^{5} - 8394 T^{6} + 2025 p T^{7} + 891 p^{2} T^{8} - 2 p^{3} T^{9} + 9 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 3 T - 24 T^{2} - 63 T^{3} + 87 T^{4} - 294 T^{5} + 1087 T^{6} - 294 p T^{7} + 87 p^{2} T^{8} - 63 p^{3} T^{9} - 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 9 T + 42 T^{2} - 47 T^{3} - 1401 T^{4} - 5028 T^{5} - 23851 T^{6} - 5028 p T^{7} - 1401 p^{2} T^{8} - 47 p^{3} T^{9} + 42 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 9 T + 3 p T^{2} - 531 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 + 12 T + 42 T^{2} + 102 T^{3} - 102 T^{4} - 16926 T^{5} - 166057 T^{6} - 16926 p T^{7} - 102 p^{2} T^{8} + 102 p^{3} T^{9} + 42 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 6 T + 45 T^{2} + 674 T^{3} + 2634 T^{4} + 17790 T^{5} + 206501 T^{6} + 17790 p T^{7} + 2634 p^{2} T^{8} + 674 p^{3} T^{9} + 45 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 9 T - 673 T^{3} - 3231 T^{4} + 10728 T^{5} + 218523 T^{6} + 10728 p T^{7} - 3231 p^{2} T^{8} - 673 p^{3} T^{9} + 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 - 3 T + 123 T^{2} - 279 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 - 3 T + 6 T^{2} + 549 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 15 T + 120 T^{2} + 621 T^{3} - 2325 T^{4} - 83550 T^{5} - 795413 T^{6} - 83550 p T^{7} - 2325 p^{2} T^{8} + 621 p^{3} T^{9} + 120 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 15 T + 3 T^{2} + 76 T^{3} + 10197 T^{4} - 27621 T^{5} - 489378 T^{6} - 27621 p T^{7} + 10197 p^{2} T^{8} + 76 p^{3} T^{9} + 3 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 9 T + 54 T^{2} + 997 T^{3} + 2313 T^{4} - 2340 T^{5} + 331203 T^{6} - 2340 p T^{7} + 2313 p^{2} T^{8} + 997 p^{3} T^{9} + 54 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 12 T - 78 T^{2} + 522 T^{3} + 14406 T^{4} - 35850 T^{5} - 971597 T^{6} - 35850 p T^{7} + 14406 p^{2} T^{8} + 522 p^{3} T^{9} - 78 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 - 30 T + 498 T^{2} - 5187 T^{3} + 498 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 + 21 T + 375 T^{2} + 3589 T^{3} + 375 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 - 24 T + 402 T^{2} - 4095 T^{3} + 402 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 9 T - 201 T^{2} + 664 T^{3} + 40017 T^{4} - 71007 T^{5} - 3746098 T^{6} - 71007 p T^{7} + 40017 p^{2} T^{8} + 664 p^{3} T^{9} - 201 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 3 T - 156 T^{2} - 555 T^{3} + 9645 T^{4} + 18822 T^{5} - 624535 T^{6} + 18822 p T^{7} + 9645 p^{2} T^{8} - 555 p^{3} T^{9} - 156 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.48887959514958448919820687881, −6.25542739924107470334920201438, −6.15389398000662034241706129998, −6.03775211735957988400157913614, −5.84045611722474095731263250603, −5.47706293428045336316904736056, −5.45010986472568899679325687920, −5.34986595818804876524319072705, −5.23478329020457030749539254421, −5.21204444987842277841942389622, −4.77985039221304692796907112016, −4.55931138472123728398412753754, −4.43076219582706695934767763619, −3.95057417499698790264056746726, −3.55022963627501722880138021158, −3.48515001727778888681026929082, −2.98094077719765026456036721239, −2.57386335007314017258916863618, −2.36773519230931644309721154281, −2.21902404294144929230327331910, −2.16695683600031131614716884452, −1.89871556772895814121356463839, −1.83054203516174676458594638970, −1.51736990681486145729814613611, −0.61015986673635251661838547009, 0.61015986673635251661838547009, 1.51736990681486145729814613611, 1.83054203516174676458594638970, 1.89871556772895814121356463839, 2.16695683600031131614716884452, 2.21902404294144929230327331910, 2.36773519230931644309721154281, 2.57386335007314017258916863618, 2.98094077719765026456036721239, 3.48515001727778888681026929082, 3.55022963627501722880138021158, 3.95057417499698790264056746726, 4.43076219582706695934767763619, 4.55931138472123728398412753754, 4.77985039221304692796907112016, 5.21204444987842277841942389622, 5.23478329020457030749539254421, 5.34986595818804876524319072705, 5.45010986472568899679325687920, 5.47706293428045336316904736056, 5.84045611722474095731263250603, 6.03775211735957988400157913614, 6.15389398000662034241706129998, 6.25542739924107470334920201438, 6.48887959514958448919820687881

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

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