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Properties

Label 12-432e6-1.1-c1e6-0-0
Degree $12$
Conductor $6.500\times 10^{15}$
Sign $1$
Analytic cond. $1684.86$
Root an. cond. $1.85729$
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  − 3·5-s − 3·7-s + 3·11-s − 12·13-s − 6·17-s − 9·19-s + 6·23-s − 9·27-s + 15·29-s + 18·31-s + 9·35-s + 15·37-s − 3·41-s + 18·43-s − 9·47-s + 15·49-s − 12·53-s − 9·55-s + 6·59-s + 18·61-s + 36·65-s + 9·67-s − 12·71-s + 3·73-s − 9·77-s − 33·79-s + 18·83-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.13·7-s + 0.904·11-s − 3.32·13-s − 1.45·17-s − 2.06·19-s + 1.25·23-s − 1.73·27-s + 2.78·29-s + 3.23·31-s + 1.52·35-s + 2.46·37-s − 0.468·41-s + 2.74·43-s − 1.31·47-s + 15/7·49-s − 1.64·53-s − 1.21·55-s + 0.781·59-s + 2.30·61-s + 4.46·65-s + 1.09·67-s − 1.42·71-s + 0.351·73-s − 1.02·77-s − 3.71·79-s + 1.97·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{18}\)
Sign: $1$
Analytic conductor: \(1684.86\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{18} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8675309851\)
\(L(\frac12)\) \(\approx\) \(0.8675309851\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p^{2} T^{3} + p^{3} T^{6} \)
good5 \( 1 + 3 T + 9 T^{2} + 9 T^{3} + 36 T^{4} + 12 T^{5} + 109 T^{6} + 12 p T^{7} + 36 p^{2} T^{8} + 9 p^{3} T^{9} + 9 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 3 T - 6 T^{2} - 50 T^{3} - 99 T^{4} + 207 T^{5} + 1401 T^{6} + 207 p T^{7} - 99 p^{2} T^{8} - 50 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T + 9 T^{2} + 9 T^{3} - 18 T^{4} + 114 T^{5} + 1225 T^{6} + 114 p T^{7} - 18 p^{2} T^{8} + 9 p^{3} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 12 T + 6 p T^{2} + 386 T^{3} + 1566 T^{4} + 5886 T^{5} + 21843 T^{6} + 5886 p T^{7} + 1566 p^{2} T^{8} + 386 p^{3} T^{9} + 6 p^{5} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 + 9 T + 36 T^{2} + 79 T^{3} - 297 T^{4} - 4806 T^{5} - 27429 T^{6} - 4806 p T^{7} - 297 p^{2} T^{8} + 79 p^{3} T^{9} + 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T + 36 T^{2} - 180 T^{3} + 1386 T^{4} - 6954 T^{5} + 33589 T^{6} - 6954 p T^{7} + 1386 p^{2} T^{8} - 180 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 15 T + 99 T^{2} - 387 T^{3} - 162 T^{4} + 17112 T^{5} - 132695 T^{6} + 17112 p T^{7} - 162 p^{2} T^{8} - 387 p^{3} T^{9} + 99 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 18 T + 171 T^{2} - 1253 T^{3} + 7263 T^{4} - 37719 T^{5} + 206634 T^{6} - 37719 p T^{7} + 7263 p^{2} T^{8} - 1253 p^{3} T^{9} + 171 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 15 T + 60 T^{2} - 289 T^{3} + 4725 T^{4} - 17730 T^{5} - 19395 T^{6} - 17730 p T^{7} + 4725 p^{2} T^{8} - 289 p^{3} T^{9} + 60 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 3 T + 36 T^{2} - 72 T^{3} + 18 p T^{4} + 1119 T^{5} + 93799 T^{6} + 1119 p T^{7} + 18 p^{3} T^{8} - 72 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 18 T + 144 T^{2} - 740 T^{3} + 432 T^{4} + 23706 T^{5} - 185739 T^{6} + 23706 p T^{7} + 432 p^{2} T^{8} - 740 p^{3} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 9 T + 9 T^{2} - 495 T^{3} - 3222 T^{4} + 3726 T^{5} + 123409 T^{6} + 3726 p T^{7} - 3222 p^{2} T^{8} - 495 p^{3} T^{9} + 9 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 6 T + 150 T^{2} + 639 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 6 T + 36 T^{2} - 261 T^{3} - 639 T^{4} + 33681 T^{5} - 161243 T^{6} + 33681 p T^{7} - 639 p^{2} T^{8} - 261 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 18 T + 153 T^{2} - 745 T^{3} - 3915 T^{4} + 107703 T^{5} - 1034862 T^{6} + 107703 p T^{7} - 3915 p^{2} T^{8} - 745 p^{3} T^{9} + 153 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 9 T + 45 T^{2} - 281 T^{3} - 1836 T^{4} + 68094 T^{5} - 564675 T^{6} + 68094 p T^{7} - 1836 p^{2} T^{8} - 281 p^{3} T^{9} + 45 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 12 T - 24 T^{2} - 738 T^{3} - 228 T^{4} + 5556 T^{5} - 117857 T^{6} + 5556 p T^{7} - 228 p^{2} T^{8} - 738 p^{3} T^{9} - 24 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 3 T - 96 T^{2} + 23 T^{3} + 2853 T^{4} + 12258 T^{5} - 46191 T^{6} + 12258 p T^{7} + 2853 p^{2} T^{8} + 23 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 33 T + 510 T^{2} + 4168 T^{3} + 3429 T^{4} - 380187 T^{5} - 5136507 T^{6} - 380187 p T^{7} + 3429 p^{2} T^{8} + 4168 p^{3} T^{9} + 510 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 18 T + 144 T^{2} - 720 T^{3} + 5580 T^{4} - 58968 T^{5} + 392545 T^{6} - 58968 p T^{7} + 5580 p^{2} T^{8} - 720 p^{3} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 15 T - 78 T^{2} - 477 T^{3} + 27177 T^{4} + 70638 T^{5} - 2238167 T^{6} + 70638 p T^{7} + 27177 p^{2} T^{8} - 477 p^{3} T^{9} - 78 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 12 T + 51 T^{2} + 1277 T^{3} + 801 T^{4} - 56169 T^{5} + 617238 T^{6} - 56169 p T^{7} + 801 p^{2} T^{8} + 1277 p^{3} T^{9} + 51 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.19680899576363822126119983740, −5.79749695874518280617044680362, −5.71886312227233640255607537408, −5.65588704367446864452499614660, −5.18434617033175072112680681030, −5.07446517026090221456821543348, −4.89076042415378985126519382349, −4.48371639236147439036421755905, −4.44110474654710692767280352875, −4.43280511971805465369517896679, −4.28621761084944313336573402595, −4.20509667891355279275319543932, −3.81190728506255511307637807657, −3.75446138765289572955166862056, −3.19876712258709981210185406307, −3.12597284972154126003472924133, −2.78424417039134136397577512886, −2.62443834784746998503013293373, −2.43684767025811834944237320512, −2.31892581321857569198198524483, −2.31605451266142563778144995912, −1.51038240137621945548132891988, −1.20866647377926384729143978350, −0.55262988847504988006474255756, −0.40895929664723854727599037438, 0.40895929664723854727599037438, 0.55262988847504988006474255756, 1.20866647377926384729143978350, 1.51038240137621945548132891988, 2.31605451266142563778144995912, 2.31892581321857569198198524483, 2.43684767025811834944237320512, 2.62443834784746998503013293373, 2.78424417039134136397577512886, 3.12597284972154126003472924133, 3.19876712258709981210185406307, 3.75446138765289572955166862056, 3.81190728506255511307637807657, 4.20509667891355279275319543932, 4.28621761084944313336573402595, 4.43280511971805465369517896679, 4.44110474654710692767280352875, 4.48371639236147439036421755905, 4.89076042415378985126519382349, 5.07446517026090221456821543348, 5.18434617033175072112680681030, 5.65588704367446864452499614660, 5.71886312227233640255607537408, 5.79749695874518280617044680362, 6.19680899576363822126119983740

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

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