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Properties

Label 12-3024e6-1.1-c1e6-0-3
Degree $12$
Conductor $7.647\times 10^{20}$
Sign $1$
Analytic cond. $1.98222\times 10^{8}$
Root an. cond. $4.91393$
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  + 5·5-s − 4·7-s − 11-s − 2·13-s + 4·17-s + 3·19-s − 7·23-s + 19·25-s + 5·29-s − 28·31-s − 20·35-s − 9·37-s + 12·41-s − 18·43-s − 6·47-s + 2·49-s − 9·53-s − 5·55-s − 8·59-s − 8·61-s − 10·65-s + 10·67-s + 14·71-s − 25·73-s + 4·77-s + 14·79-s + 8·83-s + ⋯
L(s)  = 1  + 2.23·5-s − 1.51·7-s − 0.301·11-s − 0.554·13-s + 0.970·17-s + 0.688·19-s − 1.45·23-s + 19/5·25-s + 0.928·29-s − 5.02·31-s − 3.38·35-s − 1.47·37-s + 1.87·41-s − 2.74·43-s − 0.875·47-s + 2/7·49-s − 1.23·53-s − 0.674·55-s − 1.04·59-s − 1.02·61-s − 1.24·65-s + 1.22·67-s + 1.66·71-s − 2.92·73-s + 0.455·77-s + 1.57·79-s + 0.878·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{18} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(1.98222\times 10^{8}\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8675289392\)
\(L(\frac12)\) \(\approx\) \(0.8675289392\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−4.53629784977037140641364382322, −4.42585138972749456960606413994, −4.27725348936244335791191049629, −4.27214411463124304078244557298, −3.61485649245878206821908714801, −3.52711303888128697470743459351, −3.47967061857905048601740424727, −3.47640640673052778665665831283, −3.41988914194163612634710764805, −3.28930403528794316652861590554, −3.09681998238483869647333430079, −2.80814779099609589176940875045, −2.71304172451463075834861355065, −2.37456182712221233196607365969, −2.24167582731249216918837775677, −2.17811408401637202763882551329, −2.07021849800184150426401204094, −1.89409424381947865001252320870, −1.49997061052285167798627560486, −1.35946584386771098146239441733, −1.35288735522657652698485267249, −1.31435754173717665599972443421, −0.57552616367632812342005880650, −0.44250443879081540194588754877, −0.11277964898076659454354851589, 0.11277964898076659454354851589, 0.44250443879081540194588754877, 0.57552616367632812342005880650, 1.31435754173717665599972443421, 1.35288735522657652698485267249, 1.35946584386771098146239441733, 1.49997061052285167798627560486, 1.89409424381947865001252320870, 2.07021849800184150426401204094, 2.17811408401637202763882551329, 2.24167582731249216918837775677, 2.37456182712221233196607365969, 2.71304172451463075834861355065, 2.80814779099609589176940875045, 3.09681998238483869647333430079, 3.28930403528794316652861590554, 3.41988914194163612634710764805, 3.47640640673052778665665831283, 3.47967061857905048601740424727, 3.52711303888128697470743459351, 3.61485649245878206821908714801, 4.27214411463124304078244557298, 4.27725348936244335791191049629, 4.42585138972749456960606413994, 4.53629784977037140641364382322

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

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