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Properties

Label 12-1008e6-1.1-c1e6-0-2
Degree $12$
Conductor $1.049\times 10^{18}$
Sign $1$
Analytic cond. $271910.$
Root an. cond. $2.83706$
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  − 4·3-s + 10·5-s + 2·7-s + 6·9-s − 2·11-s − 2·13-s − 40·15-s − 4·17-s + 3·19-s − 8·21-s − 14·23-s + 37·25-s − 5·27-s − 5·29-s + 14·31-s + 8·33-s + 20·35-s − 9·37-s + 8·39-s − 12·41-s − 18·43-s + 60·45-s − 3·47-s + 2·49-s + 16·51-s + 9·53-s − 20·55-s + ⋯
L(s)  = 1  − 2.30·3-s + 4.47·5-s + 0.755·7-s + 2·9-s − 0.603·11-s − 0.554·13-s − 10.3·15-s − 0.970·17-s + 0.688·19-s − 1.74·21-s − 2.91·23-s + 37/5·25-s − 0.962·27-s − 0.928·29-s + 2.51·31-s + 1.39·33-s + 3.38·35-s − 1.47·37-s + 1.28·39-s − 1.87·41-s − 2.74·43-s + 8.94·45-s − 0.437·47-s + 2/7·49-s + 2.24·51-s + 1.23·53-s − 2.69·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(271910.\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8627257114\)
\(L(\frac12)\) \(\approx\) \(0.8627257114\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + 10 T^{2} + 7 p T^{3} + 10 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 - 2 T + 2 T^{2} + 19 T^{3} + 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( ( 1 - p T + 19 T^{2} - 47 T^{3} + 19 p T^{4} - p^{3} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + T + 7 T^{2} + 5 p T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 + 2 T - 32 T^{2} - 2 p T^{3} + 730 T^{4} + 230 T^{5} - 10729 T^{6} + 230 p T^{7} + 730 p^{2} T^{8} - 2 p^{4} T^{9} - 32 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T + 9 T^{2} + 92 T^{3} + 58 T^{4} - 20 T^{5} + 5393 T^{6} - 20 p T^{7} + 58 p^{2} T^{8} + 92 p^{3} T^{9} + 9 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 42 T^{2} + 61 T^{3} + 69 p T^{4} - 726 T^{5} - 27501 T^{6} - 726 p T^{7} + 69 p^{3} T^{8} + 61 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 + 7 T + 73 T^{2} + 319 T^{3} + 73 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 + 5 T - 30 T^{2} - 371 T^{3} - 185 T^{4} + 6020 T^{5} + 44357 T^{6} + 6020 p T^{7} - 185 p^{2} T^{8} - 371 p^{3} T^{9} - 30 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 14 T + 58 T^{2} - 250 T^{3} + 2992 T^{4} - 9728 T^{5} - 11857 T^{6} - 9728 p T^{7} + 2992 p^{2} T^{8} - 250 p^{3} T^{9} + 58 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 4875 p T^{7} + 1293 p^{2} T^{8} - 268 p^{3} T^{9} - 21 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T - 18 T^{2} - 78 T^{3} + 7470 T^{4} + 24546 T^{5} - 158105 T^{6} + 24546 p T^{7} + 7470 p^{2} T^{8} - 78 p^{3} T^{9} - 18 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 18 T + 114 T^{2} + 682 T^{3} + 7188 T^{4} + 33492 T^{5} + 63039 T^{6} + 33492 p T^{7} + 7188 p^{2} T^{8} + 682 p^{3} T^{9} + 114 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 3 T - 108 T^{2} - 267 T^{3} + 7263 T^{4} + 9786 T^{5} - 360137 T^{6} + 9786 p T^{7} + 7263 p^{2} T^{8} - 267 p^{3} T^{9} - 108 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 9 T - 36 T^{2} + 873 T^{3} - 1179 T^{4} - 26334 T^{5} + 272077 T^{6} - 26334 p T^{7} - 1179 p^{2} T^{8} + 873 p^{3} T^{9} - 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 4 T - 60 T^{2} - 994 T^{3} - 1304 T^{4} + 464 p T^{5} + 7381 p T^{6} + 464 p^{2} T^{7} - 1304 p^{2} T^{8} - 994 p^{3} T^{9} - 60 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 4 T - 32 T^{2} - 650 T^{3} + 292 T^{4} + 19532 T^{5} + 306323 T^{6} + 19532 p T^{7} + 292 p^{2} T^{8} - 650 p^{3} T^{9} - 32 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 5 T - 118 T^{2} - 327 T^{3} + 8263 T^{4} + 1138 T^{5} - 609341 T^{6} + 1138 p T^{7} + 8263 p^{2} T^{8} - 327 p^{3} T^{9} - 118 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 7 T + 163 T^{2} + 895 T^{3} + 163 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 115046 p T^{7} + 20533 p^{2} T^{8} + 2073 p^{3} T^{9} + 254 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 7 T - 44 T^{2} - 19 T^{3} - 1043 T^{4} - 28016 T^{5} + 109223 T^{6} - 28016 p T^{7} - 1043 p^{2} T^{8} - 19 p^{3} T^{9} - 44 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 8 T - 180 T^{2} - 518 T^{3} + 29404 T^{4} + 32420 T^{5} - 2713585 T^{6} + 32420 p T^{7} + 29404 p^{2} T^{8} - 518 p^{3} T^{9} - 180 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 9 T - 180 T^{2} - 729 T^{3} + 31041 T^{4} + 54846 T^{5} - 2925911 T^{6} + 54846 p T^{7} + 31041 p^{2} T^{8} - 729 p^{3} T^{9} - 180 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 28 T + 257 T^{2} + 2820 T^{3} + 59506 T^{4} + 545924 T^{5} + 3126001 T^{6} + 545924 p T^{7} + 59506 p^{2} T^{8} + 2820 p^{3} T^{9} + 257 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.33180840133492559648434901668, −5.19607774219170389925403083351, −5.16061646378169559277793887487, −5.15905943986277968099351958805, −4.68701181304165230751247305788, −4.60723857661338138857017007134, −4.42936745977076228009183857697, −4.20748437566498736808595903149, −4.01518092750174708105673252263, −3.94543305387240282319054500557, −3.77606647391854451661155897704, −3.12480129779635462651186558067, −3.10029542898103028857316454742, −2.96648913038438996129268562835, −2.74724866109442533872490762208, −2.66538709856972608246273600499, −1.98945166766857644454963637492, −1.98521449318396799475856611423, −1.98406000552015288990483591268, −1.80679661295125280428201686411, −1.73673665905131937270034253148, −1.46888033376599572451699296229, −1.17688736407831848937463953416, −0.42949306127341024110573346703, −0.21725722712362141910799107253, 0.21725722712362141910799107253, 0.42949306127341024110573346703, 1.17688736407831848937463953416, 1.46888033376599572451699296229, 1.73673665905131937270034253148, 1.80679661295125280428201686411, 1.98406000552015288990483591268, 1.98521449318396799475856611423, 1.98945166766857644454963637492, 2.66538709856972608246273600499, 2.74724866109442533872490762208, 2.96648913038438996129268562835, 3.10029542898103028857316454742, 3.12480129779635462651186558067, 3.77606647391854451661155897704, 3.94543305387240282319054500557, 4.01518092750174708105673252263, 4.20748437566498736808595903149, 4.42936745977076228009183857697, 4.60723857661338138857017007134, 4.68701181304165230751247305788, 5.15905943986277968099351958805, 5.16061646378169559277793887487, 5.19607774219170389925403083351, 5.33180840133492559648434901668

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

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