Properties

Label 12-3e36-1.1-c1e6-0-8
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 $0$

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: \(0\)
Selberg data: \((12,\ 3^{36} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(33.61776149\)
\(L(\frac12)\) \(\approx\) \(33.61776149\)
\(L(\frac{3}{2})\) 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

−5.56285365207551545232442198576, −5.18499163173812978906141241042, −5.06803651250874245538152842376, −5.04755137468847749112526420927, −5.02188312889748839748182524500, −4.82369108770828682500824239274, −4.68307687198509270075605446473, −4.10512819238575549410036105488, −4.02929669230595519511994488678, −4.01869655240563557747779861066, −3.94253613764637251408594541676, −3.88994231302576247410678738790, −3.71568342102786692878602136320, −3.22355982813605523705427438329, −3.16646500836920234526378260300, −2.87142918998076205608054637174, −2.76093908222073722224892930864, −2.42874296961364628202601194987, −2.31286785643988274643568642026, −1.81368532216312820667387991120, −1.73194096264220742057972577275, −1.36316394305759133768250012775, −1.10259637188106532316593793706, −0.893855383034941114715143574181, −0.838237704166559149232616522113, 0.838237704166559149232616522113, 0.893855383034941114715143574181, 1.10259637188106532316593793706, 1.36316394305759133768250012775, 1.73194096264220742057972577275, 1.81368532216312820667387991120, 2.31286785643988274643568642026, 2.42874296961364628202601194987, 2.76093908222073722224892930864, 2.87142918998076205608054637174, 3.16646500836920234526378260300, 3.22355982813605523705427438329, 3.71568342102786692878602136320, 3.88994231302576247410678738790, 3.94253613764637251408594541676, 4.01869655240563557747779861066, 4.02929669230595519511994488678, 4.10512819238575549410036105488, 4.68307687198509270075605446473, 4.82369108770828682500824239274, 5.02188312889748839748182524500, 5.04755137468847749112526420927, 5.06803651250874245538152842376, 5.18499163173812978906141241042, 5.56285365207551545232442198576

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

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