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Properties

Label 12-19e6-1.1-c1e6-0-0
Degree $12$
Conductor $47045881$
Sign $1$
Analytic cond. $1.21950\times 10^{-5}$
Root an. cond. $0.389507$
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  − 6·2-s − 3·3-s + 18·4-s − 6·5-s + 18·6-s − 36·8-s + 6·9-s + 36·10-s − 54·12-s − 3·13-s + 18·15-s + 54·16-s + 3·17-s − 36·18-s − 12·19-s − 108·20-s + 6·23-s + 108·24-s + 18·25-s + 18·26-s − 10·27-s − 3·29-s − 108·30-s + 9·31-s − 69·32-s − 18·34-s + 108·36-s + ⋯
L(s)  = 1  − 4.24·2-s − 1.73·3-s + 9·4-s − 2.68·5-s + 7.34·6-s − 12.7·8-s + 2·9-s + 11.3·10-s − 15.5·12-s − 0.832·13-s + 4.64·15-s + 27/2·16-s + 0.727·17-s − 8.48·18-s − 2.75·19-s − 24.1·20-s + 1.25·23-s + 22.0·24-s + 18/5·25-s + 3.53·26-s − 1.92·27-s − 0.557·29-s − 19.7·30-s + 1.61·31-s − 12.1·32-s − 3.08·34-s + 18·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47045881 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47045881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(47045881\)    =    \(19^{6}\)
Sign: $1$
Analytic conductor: \(1.21950\times 10^{-5}\)
Root analytic conductor: \(0.389507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 47045881,\ (\ :[1/2]^{6}),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.004687604950\)
\(L(\frac12)\) \(\approx\) \(0.004687604950\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + 12 T + 78 T^{2} + 385 T^{3} + 78 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 + 3 p T + 9 p T^{2} + 9 p^{2} T^{3} + 27 p T^{4} + 69 T^{5} + 91 T^{6} + 69 p T^{7} + 27 p^{3} T^{8} + 9 p^{5} T^{9} + 9 p^{5} T^{10} + 3 p^{6} T^{11} + p^{6} T^{12} \)
3 \( 1 + p T + p T^{2} + T^{3} - 4 p T^{4} - 10 p T^{5} - 35 T^{6} - 10 p^{2} T^{7} - 4 p^{3} T^{8} + p^{3} T^{9} + p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
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 \( 1 - 18 T^{2} + 2 T^{3} + 198 T^{4} - 18 T^{5} - 1581 T^{6} - 18 p T^{7} + 198 p^{2} T^{8} + 2 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 24 T^{2} - 18 T^{3} + 312 T^{4} + 216 T^{5} - 3593 T^{6} + 216 p T^{7} + 312 p^{2} T^{8} - 18 p^{3} T^{9} - 24 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 3 T + 24 T^{2} + 2 p T^{3} + 315 T^{4} + 261 T^{5} + 4905 T^{6} + 261 p T^{7} + 315 p^{2} T^{8} + 2 p^{4} T^{9} + 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 3 T - 72 T^{3} - 117 T^{4} + 75 p T^{5} + 1369 T^{6} + 75 p^{2} T^{7} - 117 p^{2} T^{8} - 72 p^{3} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T + 36 T^{2} - 54 T^{3} + 576 T^{4} + 516 T^{5} + 4969 T^{6} + 516 p T^{7} + 576 p^{2} T^{8} - 54 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 36 T^{2} + 378 T^{3} + 1872 T^{4} + 9921 T^{5} + 94159 T^{6} + 9921 p T^{7} + 1872 p^{2} T^{8} + 378 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 9 T - 18 T^{2} + 119 T^{3} + 2187 T^{4} - 3402 T^{5} - 67065 T^{6} - 3402 p T^{7} + 2187 p^{2} T^{8} + 119 p^{3} T^{9} - 18 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 90 T^{2} - 17 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \)
41 \( 1 - 21 T + 162 T^{2} - 180 T^{3} - 4707 T^{4} + 28401 T^{5} - 103463 T^{6} + 28401 p T^{7} - 4707 p^{2} T^{8} - 180 p^{3} T^{9} + 162 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T + 60 T^{2} + 8 T^{3} + 2691 T^{4} + 2799 T^{5} + 147141 T^{6} + 2799 p T^{7} + 2691 p^{2} T^{8} + 8 p^{3} T^{9} + 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 3 T + 54 T^{2} + 306 T^{3} + 4635 T^{4} + 537 p T^{5} + 190171 T^{6} + 537 p^{2} T^{7} + 4635 p^{2} T^{8} + 306 p^{3} T^{9} + 54 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 3 T + 234 T^{3} - 1521 T^{4} - 27771 T^{5} - 29411 T^{6} - 27771 p T^{7} - 1521 p^{2} T^{8} + 234 p^{3} T^{9} + 3 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 12 T + 18 T^{2} + 1080 T^{3} - 7614 T^{4} - 29982 T^{5} + 754273 T^{6} - 29982 p T^{7} - 7614 p^{2} T^{8} + 1080 p^{3} T^{9} + 18 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 24 T^{2} - 586 T^{3} - 4140 T^{4} + 44676 T^{5} + 736335 T^{6} + 44676 p T^{7} - 4140 p^{2} T^{8} - 586 p^{3} T^{9} + 24 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 30 T + 348 T^{2} + 1322 T^{3} - 6408 T^{4} - 55008 T^{5} - 101691 T^{6} - 55008 p T^{7} - 6408 p^{2} T^{8} + 1322 p^{3} T^{9} + 348 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 6 T - 36 T^{2} + 594 T^{3} + 3240 T^{4} + 14892 T^{5} + 665785 T^{6} + 14892 p T^{7} + 3240 p^{2} T^{8} + 594 p^{3} T^{9} - 36 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 12 T + 96 T^{2} + 512 T^{3} + 7776 T^{4} + 102924 T^{5} + 981639 T^{6} + 102924 p T^{7} + 7776 p^{2} T^{8} + 512 p^{3} T^{9} + 96 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 39 T + 708 T^{2} + 8198 T^{3} + 67068 T^{4} + 403623 T^{5} + 2596617 T^{6} + 403623 p T^{7} + 67068 p^{2} T^{8} + 8198 p^{3} T^{9} + 708 p^{4} T^{10} + 39 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 60 T^{2} + 918 T^{3} - 1380 T^{4} - 27540 T^{5} + 1055455 T^{6} - 27540 p T^{7} - 1380 p^{2} T^{8} + 918 p^{3} T^{9} - 60 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 + 12 T + 54 T^{2} + 1035 T^{3} - 279 T^{4} - 69891 T^{5} + 3961 T^{6} - 69891 p T^{7} - 279 p^{2} T^{8} + 1035 p^{3} T^{9} + 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 18 T + 234 T^{2} - 3310 T^{3} + 39204 T^{4} - 371520 T^{5} + 3748107 T^{6} - 371520 p T^{7} + 39204 p^{2} T^{8} - 3310 p^{3} T^{9} + 234 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
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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

−11.20109023371604474172051081790, −11.01900410709222093023054735482, −10.63686474955047339161046226295, −10.44393828019207959773495584508, −10.26542560881370856685877618346, −10.16429174854356309503426923748, −9.789451133347758811425143701092, −9.394457177278772894493182185997, −8.964246542321939847993066642946, −8.893438845980321839455126817252, −8.755142049334680624551706331763, −8.425980951481141939577225884869, −8.154055427497165739338416140603, −7.60731457672383664023175471594, −7.60518620624050872929393087320, −7.39209480653299014951293223245, −7.17829429165426303949443463935, −6.84536631178454262481636256613, −6.33180420818185583642667808859, −5.79418956427122526140543409541, −5.78031012960520657860516432631, −4.62462035195005422634167108817, −4.34529178975668977150684737643, −4.23429636524392747738706203360, −2.95600261473892036517366115513, 2.95600261473892036517366115513, 4.23429636524392747738706203360, 4.34529178975668977150684737643, 4.62462035195005422634167108817, 5.78031012960520657860516432631, 5.79418956427122526140543409541, 6.33180420818185583642667808859, 6.84536631178454262481636256613, 7.17829429165426303949443463935, 7.39209480653299014951293223245, 7.60518620624050872929393087320, 7.60731457672383664023175471594, 8.154055427497165739338416140603, 8.425980951481141939577225884869, 8.755142049334680624551706331763, 8.893438845980321839455126817252, 8.964246542321939847993066642946, 9.394457177278772894493182185997, 9.789451133347758811425143701092, 10.16429174854356309503426923748, 10.26542560881370856685877618346, 10.44393828019207959773495584508, 10.63686474955047339161046226295, 11.01900410709222093023054735482, 11.20109023371604474172051081790

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

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