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Properties

Label 12-1323e6-1.1-c1e6-0-6
Degree $12$
Conductor $5.362\times 10^{18}$
Sign $1$
Analytic cond. $1.39002\times 10^{6}$
Root an. cond. $3.25026$
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  − 6·2-s + 15·4-s + 3·5-s − 18·8-s − 18·10-s + 6·11-s + 3·13-s + 3·16-s + 6·17-s + 3·19-s + 45·20-s − 36·22-s + 12·23-s + 15·25-s − 18·26-s + 9·29-s − 6·31-s + 30·32-s − 36·34-s + 3·37-s − 18·38-s − 54·40-s + 3·43-s + 90·44-s − 72·46-s − 6·47-s − 90·50-s + ⋯
L(s)  = 1  − 4.24·2-s + 15/2·4-s + 1.34·5-s − 6.36·8-s − 5.69·10-s + 1.80·11-s + 0.832·13-s + 3/4·16-s + 1.45·17-s + 0.688·19-s + 10.0·20-s − 7.67·22-s + 2.50·23-s + 3·25-s − 3.53·26-s + 1.67·29-s − 1.07·31-s + 5.30·32-s − 6.17·34-s + 0.493·37-s − 2.91·38-s − 8.53·40-s + 0.457·43-s + 13.5·44-s − 10.6·46-s − 0.875·47-s − 12.7·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.39002\times 10^{6}\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6859937569\)
\(L(\frac12)\) \(\approx\) \(0.6859937569\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( ( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
5 \( 1 - 3 T - 6 T^{2} + 9 T^{3} + 69 T^{4} - 6 p T^{5} - 371 T^{6} - 6 p^{2} T^{7} + 69 p^{2} T^{8} + 9 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 6 T - 6 T^{2} + 18 T^{3} + 492 T^{4} - 852 T^{5} - 2873 T^{6} - 852 p T^{7} + 492 p^{2} T^{8} + 18 p^{3} T^{9} - 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 135 p T^{7} + 45 p^{2} T^{8} - 76 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 6 T - 24 T^{2} + 54 T^{3} + 1338 T^{4} - 1914 T^{5} - 18929 T^{6} - 1914 p T^{7} + 1338 p^{2} T^{8} + 54 p^{3} T^{9} - 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 42 T^{2} + 41 T^{3} + 1341 T^{4} - 216 T^{5} - 29541 T^{6} - 216 p T^{7} + 1341 p^{2} T^{8} + 41 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 12 T + 48 T^{2} - 54 T^{3} + 420 T^{4} - 6060 T^{5} + 37591 T^{6} - 6060 p T^{7} + 420 p^{2} T^{8} - 54 p^{3} T^{9} + 48 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 9 T + 30 T^{2} - 81 T^{3} - 579 T^{4} + 9414 T^{5} - 59051 T^{6} + 9414 p T^{7} - 579 p^{2} T^{8} - 81 p^{3} T^{9} + 30 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 3 T + 15 T^{2} - 137 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 3 T - 24 T^{2} - 301 T^{3} + 171 T^{4} + 6552 T^{5} + 58893 T^{6} + 6552 p T^{7} + 171 p^{2} T^{8} - 301 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 114 T^{2} + 18 T^{3} + 8322 T^{4} - 1026 T^{5} - 394913 T^{6} - 1026 p T^{7} + 8322 p^{2} T^{8} + 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 5670 p T^{7} + 9063 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 3 T + 87 T^{2} + 333 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 - 6 T - 114 T^{2} + 378 T^{3} + 10716 T^{4} - 17304 T^{5} - 587549 T^{6} - 17304 p T^{7} + 10716 p^{2} T^{8} + 378 p^{3} T^{9} - 114 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 - 3 T + 105 T^{2} - 405 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 6 T + 168 T^{2} - 713 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 + 12 T + 222 T^{2} + 1591 T^{3} + 222 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 159 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 21 T + 138 T^{2} - 769 T^{3} + 10953 T^{4} - 30402 T^{5} - 450903 T^{6} - 30402 p T^{7} + 10953 p^{2} T^{8} - 769 p^{3} T^{9} + 138 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 + 21 T + 357 T^{2} + 3499 T^{3} + 357 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 + 18 T + 30 T^{2} - 702 T^{3} + 8088 T^{4} + 126648 T^{5} + 719359 T^{6} + 126648 p T^{7} + 8088 p^{2} T^{8} - 702 p^{3} T^{9} + 30 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 12 T - 60 T^{2} + 198 T^{3} + 7584 T^{4} + 70800 T^{5} - 1684181 T^{6} + 70800 p T^{7} + 7584 p^{2} T^{8} + 198 p^{3} T^{9} - 60 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 11502 p T^{7} + 2421 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.15647210108345494806655841704, −4.86380285581696211941548942749, −4.79542064840869391124347659342, −4.67000187049022545446688180544, −4.53272063276787728897164247088, −4.37733122918936761059266255226, −3.93292738038095162567008306760, −3.84639525545306640528622805239, −3.75782377640479351629457596012, −3.54060668688092052242506071789, −3.37981372863964212290991161340, −2.95406513760523355890324987096, −2.89097028496577492197810740024, −2.88553098963996835285390837704, −2.53019707865550335147930614360, −2.49642580341731286473168079976, −2.21192925779734216232361606587, −1.50691906677424877124384919067, −1.49376423775150891738678551928, −1.47004014590917633133735004636, −1.30529625134466964710113738245, −1.03744470626005648041139332905, −0.974039639536031322478803260247, −0.48897064206948034199398703299, −0.43651933964689686108467198923, 0.43651933964689686108467198923, 0.48897064206948034199398703299, 0.974039639536031322478803260247, 1.03744470626005648041139332905, 1.30529625134466964710113738245, 1.47004014590917633133735004636, 1.49376423775150891738678551928, 1.50691906677424877124384919067, 2.21192925779734216232361606587, 2.49642580341731286473168079976, 2.53019707865550335147930614360, 2.88553098963996835285390837704, 2.89097028496577492197810740024, 2.95406513760523355890324987096, 3.37981372863964212290991161340, 3.54060668688092052242506071789, 3.75782377640479351629457596012, 3.84639525545306640528622805239, 3.93292738038095162567008306760, 4.37733122918936761059266255226, 4.53272063276787728897164247088, 4.67000187049022545446688180544, 4.79542064840869391124347659342, 4.86380285581696211941548942749, 5.15647210108345494806655841704

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

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