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Properties

Label 12-1008e6-1.1-c1e6-0-5
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5·5-s − 4·7-s + 11-s − 2·13-s − 10·15-s − 4·17-s + 3·19-s − 8·21-s + 7·23-s + 19·25-s − 5·27-s − 5·29-s − 28·31-s + 2·33-s + 20·35-s − 9·37-s − 4·39-s − 12·41-s − 18·43-s + 6·47-s + 2·49-s − 8·51-s + 9·53-s − 5·55-s + 6·57-s + 8·59-s + ⋯
L(s)  = 1  + 1.15·3-s − 2.23·5-s − 1.51·7-s + 0.301·11-s − 0.554·13-s − 2.58·15-s − 0.970·17-s + 0.688·19-s − 1.74·21-s + 1.45·23-s + 19/5·25-s − 0.962·27-s − 0.928·29-s − 5.02·31-s + 0.348·33-s + 3.38·35-s − 1.47·37-s − 0.640·39-s − 1.87·41-s − 2.74·43-s + 0.875·47-s + 2/7·49-s − 1.12·51-s + 1.23·53-s − 0.674·55-s + 0.794·57-s + 1.04·59-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

\(L(1)\) \(\approx\) \(1.126825419\)
\(L(\frac12)\) \(\approx\) \(1.126825419\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + 4 T^{2} - p T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 + 4 T + 2 p T^{2} + 55 T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + p T + 6 T^{2} + T^{3} + 31 T^{4} + 68 T^{5} + 29 T^{6} + 68 p T^{7} + 31 p^{2} T^{8} + p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 - T - 6 T^{2} + 103 T^{3} - 83 T^{4} - 32 p T^{5} + 457 p T^{6} - 32 p^{2} T^{7} - 83 p^{2} T^{8} + 103 p^{3} T^{9} - 6 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 7 T - 24 T^{2} + 127 T^{3} + 1417 T^{4} - 3484 T^{5} - 22393 T^{6} - 3484 p T^{7} + 1417 p^{2} T^{8} + 127 p^{3} T^{9} - 24 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
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 \( ( 1 + 14 T + 138 T^{2} + 841 T^{3} + 138 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 3 T + 117 T^{2} - 309 T^{3} + 117 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 4 T + 76 T^{2} - 11 p T^{3} + 76 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 4 T + 48 T^{2} - 229 T^{3} + 48 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - 5 T + 143 T^{2} - 521 T^{3} + 143 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 7 T + 93 T^{2} - 335 T^{3} + 93 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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

−5.34905806017071339235062978138, −5.33002620918096334115765300154, −4.84408498756668304754714233525, −4.82009077992418943764142267683, −4.49775000284310319760246880895, −4.28111835877467987750646258256, −4.21034271862034242125125468928, −3.95875791062765499669106217752, −3.92744546784520735714179940575, −3.88964880745718821979782110897, −3.47225628583786541125146289250, −3.24081505021332809111243316001, −3.11528963097026558462445110712, −3.07909097557052223190310110335, −3.04887256580258220680552017370, −3.02278571963472625523687047556, −2.68894725055064871122308063923, −2.10317197255407508664578916017, −1.91767305293625226963001913516, −1.71477585194065126356751677262, −1.68659109966072431110295345329, −1.60591048617870841063808884036, −0.62034208852490406629821499416, −0.41518756409511678105229080143, −0.35521437908048923609859712958, 0.35521437908048923609859712958, 0.41518756409511678105229080143, 0.62034208852490406629821499416, 1.60591048617870841063808884036, 1.68659109966072431110295345329, 1.71477585194065126356751677262, 1.91767305293625226963001913516, 2.10317197255407508664578916017, 2.68894725055064871122308063923, 3.02278571963472625523687047556, 3.04887256580258220680552017370, 3.07909097557052223190310110335, 3.11528963097026558462445110712, 3.24081505021332809111243316001, 3.47225628583786541125146289250, 3.88964880745718821979782110897, 3.92744546784520735714179940575, 3.95875791062765499669106217752, 4.21034271862034242125125468928, 4.28111835877467987750646258256, 4.49775000284310319760246880895, 4.82009077992418943764142267683, 4.84408498756668304754714233525, 5.33002620918096334115765300154, 5.34905806017071339235062978138

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

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