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Properties

Label 12-3e36-1.1-c1e6-0-12
Degree $12$
Conductor $1.501\times 10^{17}$
Sign $1$
Analytic cond. $38907.0$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 6·5-s + 9·8-s + 18·10-s − 12·11-s − 9·16-s − 9·17-s + 3·19-s + 36·22-s − 15·23-s − 12·29-s − 3·32-s + 27·34-s + 3·37-s − 9·38-s − 54·40-s − 15·41-s + 45·46-s − 21·47-s − 27·49-s − 9·53-s + 72·55-s + 36·58-s − 24·59-s − 9·61-s + 8·64-s − 9·67-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.68·5-s + 3.18·8-s + 5.69·10-s − 3.61·11-s − 9/4·16-s − 2.18·17-s + 0.688·19-s + 7.67·22-s − 3.12·23-s − 2.22·29-s − 0.530·32-s + 4.63·34-s + 0.493·37-s − 1.45·38-s − 8.53·40-s − 2.34·41-s + 6.63·46-s − 3.06·47-s − 3.85·49-s − 1.23·53-s + 9.70·55-s + 4.72·58-s − 3.12·59-s − 1.15·61-s + 64-s − 1.09·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{36}\)
Sign: $1$
Analytic conductor: \(38907.0\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{36} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + 3 T + 9 T^{2} + 9 p T^{3} + 9 p^{2} T^{4} + 57 T^{5} + 91 T^{6} + 57 p T^{7} + 9 p^{4} T^{8} + 9 p^{4} T^{9} + 9 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 6 T + 36 T^{2} + 126 T^{3} + 441 T^{4} + 1113 T^{5} + 2863 T^{6} + 1113 p T^{7} + 441 p^{2} T^{8} + 126 p^{3} T^{9} + 36 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 27 T^{2} + 11 T^{3} + 351 T^{4} + 216 T^{5} + 2937 T^{6} + 216 p T^{7} + 351 p^{2} T^{8} + 11 p^{3} T^{9} + 27 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 + 12 T + 117 T^{2} + 756 T^{3} + 4140 T^{4} + 17715 T^{5} + 65431 T^{6} + 17715 p T^{7} + 4140 p^{2} T^{8} + 756 p^{3} T^{9} + 117 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 54 T^{2} + 2 T^{3} + 1377 T^{4} + 135 T^{5} + 21945 T^{6} + 135 p T^{7} + 1377 p^{2} T^{8} + 2 p^{3} T^{9} + 54 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 9 T + 111 T^{2} + 711 T^{3} + 4893 T^{4} + 23337 T^{5} + 112057 T^{6} + 23337 p T^{7} + 4893 p^{2} T^{8} + 711 p^{3} T^{9} + 111 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T + 84 T^{2} - 13 p T^{3} + 3303 T^{4} - 8784 T^{5} + 4137 p T^{6} - 8784 p T^{7} + 3303 p^{2} T^{8} - 13 p^{4} T^{9} + 84 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 15 T + 198 T^{2} + 1692 T^{3} + 13023 T^{4} + 77028 T^{5} + 414235 T^{6} + 77028 p T^{7} + 13023 p^{2} T^{8} + 1692 p^{3} T^{9} + 198 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 12 T + 171 T^{2} + 1278 T^{3} + 369 p T^{4} + 59628 T^{5} + 381601 T^{6} + 59628 p T^{7} + 369 p^{3} T^{8} + 1278 p^{3} T^{9} + 171 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 135 T^{2} + 191 T^{3} + 7911 T^{4} + 17658 T^{5} + 290757 T^{6} + 17658 p T^{7} + 7911 p^{2} T^{8} + 191 p^{3} T^{9} + 135 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 3 T + 165 T^{2} - 301 T^{3} + 12591 T^{4} - 16749 T^{5} + 586203 T^{6} - 16749 p T^{7} + 12591 p^{2} T^{8} - 301 p^{3} T^{9} + 165 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 15 T + 252 T^{2} + 2475 T^{3} + 24849 T^{4} + 180636 T^{5} + 1331587 T^{6} + 180636 p T^{7} + 24849 p^{2} T^{8} + 2475 p^{3} T^{9} + 252 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 162 T^{2} + 173 T^{3} + 13581 T^{4} + 14715 T^{5} + 729723 T^{6} + 14715 p T^{7} + 13581 p^{2} T^{8} + 173 p^{3} T^{9} + 162 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 + 21 T + 387 T^{2} + 4707 T^{3} + 50769 T^{4} + 431463 T^{5} + 3276703 T^{6} + 431463 p T^{7} + 50769 p^{2} T^{8} + 4707 p^{3} T^{9} + 387 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 9 T + 210 T^{2} + 1872 T^{3} + 23856 T^{4} + 168327 T^{5} + 1634317 T^{6} + 168327 p T^{7} + 23856 p^{2} T^{8} + 1872 p^{3} T^{9} + 210 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 24 T + 495 T^{2} + 6651 T^{3} + 79659 T^{4} + 743550 T^{5} + 6351049 T^{6} + 743550 p T^{7} + 79659 p^{2} T^{8} + 6651 p^{3} T^{9} + 495 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 9 T + 207 T^{2} + 2225 T^{3} + 26217 T^{4} + 214785 T^{5} + 2128485 T^{6} + 214785 p T^{7} + 26217 p^{2} T^{8} + 2225 p^{3} T^{9} + 207 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 9 T + 288 T^{2} + 2081 T^{3} + 37935 T^{4} + 219402 T^{5} + 3096273 T^{6} + 219402 p T^{7} + 37935 p^{2} T^{8} + 2081 p^{3} T^{9} + 288 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 27 T + 651 T^{2} + 10071 T^{3} + 138813 T^{4} + 1464345 T^{5} + 13863913 T^{6} + 1464345 p T^{7} + 138813 p^{2} T^{8} + 10071 p^{3} T^{9} + 651 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 6 T + 264 T^{2} + 1940 T^{3} + 33111 T^{4} + 266427 T^{5} + 2798097 T^{6} + 266427 p T^{7} + 33111 p^{2} T^{8} + 1940 p^{3} T^{9} + 264 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 297 T^{2} - 70 T^{3} + 42768 T^{4} - 22869 T^{5} + 4038141 T^{6} - 22869 p T^{7} + 42768 p^{2} T^{8} - 70 p^{3} T^{9} + 297 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 + 12 T + 333 T^{2} + 3825 T^{3} + 58977 T^{4} + 542334 T^{5} + 6262309 T^{6} + 542334 p T^{7} + 58977 p^{2} T^{8} + 3825 p^{3} T^{9} + 333 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 9 T + 354 T^{2} + 2979 T^{3} + 59703 T^{4} + 469404 T^{5} + 6461593 T^{6} + 469404 p T^{7} + 59703 p^{2} T^{8} + 2979 p^{3} T^{9} + 354 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 378 T^{2} + 713 T^{3} + 67581 T^{4} + 185517 T^{5} + 7814685 T^{6} + 185517 p T^{7} + 67581 p^{2} T^{8} + 713 p^{3} T^{9} + 378 p^{4} T^{10} + p^{6} T^{12} \)
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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

−6.08290033506018708127006184561, −5.95809874712577181210718782332, −5.53440045323573199354654852474, −5.53051695146413229094843262166, −5.42684227220377653042652778140, −5.23447279546781579907709607211, −4.93136339462419425007243312248, −4.70689890497037420915147465874, −4.63087450589062558570813055946, −4.35960612079513948442785350673, −4.35498958548368895676105017199, −4.33997754212319295297559547938, −4.24543471662823034700266946181, −3.61316423255547458311858525856, −3.58707604508317782956021549136, −3.52951825264974116409177435744, −3.23478917790178859565313368717, −3.11748094666312729388666379753, −2.82521821242815915221485454620, −2.62881899175052538848358130059, −2.27617849466440400491199217303, −1.94560220139732442041610272342, −1.64566082767815986874265981994, −1.64306494038862216364660489403, −1.49984527792155789246925648260, 0, 0, 0, 0, 0, 0, 1.49984527792155789246925648260, 1.64306494038862216364660489403, 1.64566082767815986874265981994, 1.94560220139732442041610272342, 2.27617849466440400491199217303, 2.62881899175052538848358130059, 2.82521821242815915221485454620, 3.11748094666312729388666379753, 3.23478917790178859565313368717, 3.52951825264974116409177435744, 3.58707604508317782956021549136, 3.61316423255547458311858525856, 4.24543471662823034700266946181, 4.33997754212319295297559547938, 4.35498958548368895676105017199, 4.35960612079513948442785350673, 4.63087450589062558570813055946, 4.70689890497037420915147465874, 4.93136339462419425007243312248, 5.23447279546781579907709607211, 5.42684227220377653042652778140, 5.53051695146413229094843262166, 5.53440045323573199354654852474, 5.95809874712577181210718782332, 6.08290033506018708127006184561

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

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