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Properties

Label 12-3479e6-1.1-c0e6-0-0
Degree $12$
Conductor $1.773\times 10^{21}$
Sign $1$
Analytic cond. $27.3948$
Root an. cond. $1.31766$
Motivic weight $0$
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  − 2·2-s + 4-s + 9-s − 2·18-s − 6·25-s − 5·29-s + 36-s + 2·37-s + 5·43-s + 12·50-s + 10·58-s − 71-s − 4·74-s + 2·79-s − 10·86-s − 6·100-s + 2·107-s + 5·109-s − 5·116-s − 121-s + 127-s + 2·128-s + 131-s + 137-s + 139-s + 2·142-s + 2·148-s + ⋯
L(s)  = 1  − 2·2-s + 4-s + 9-s − 2·18-s − 6·25-s − 5·29-s + 36-s + 2·37-s + 5·43-s + 12·50-s + 10·58-s − 71-s − 4·74-s + 2·79-s − 10·86-s − 6·100-s + 2·107-s + 5·109-s − 5·116-s − 121-s + 127-s + 2·128-s + 131-s + 137-s + 139-s + 2·142-s + 2·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 71^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 71^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 71^{6}\)
Sign: $1$
Analytic conductor: \(27.3948\)
Root analytic conductor: \(1.31766\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{12} \cdot 71^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05239149162\)
\(L(\frac12)\) \(\approx\) \(0.05239149162\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
71 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
good2 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
5 \( ( 1 + T^{2} )^{6} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
17 \( ( 1 - T )^{6}( 1 + T )^{6} \)
19 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
29 \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
43 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
73 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
83 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
89 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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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

−4.41860805657770448265467793288, −4.40989406108100661560738473135, −4.24816952159144144917680220250, −4.11100859110189924859848989142, −3.97299059955552216557339652913, −3.92340205981661260774166535316, −3.91568811073459788878805449103, −3.71646914728754141322944563918, −3.44400305551050215064272721348, −3.40677883811771216504954886130, −3.23826428875752414995723576962, −3.04089691281666289899115726355, −2.69193728838298276799641808520, −2.42212037504017540819117806438, −2.38375473016090557588198345211, −2.19390987259617518107973245289, −2.11258337538148573950512432305, −2.05810892733831576644429623908, −1.73178095800492796266937119658, −1.68999589125439346804391166113, −1.39861911881030270543555527029, −1.17484247657792286619657792801, −0.74143931724625328950622531049, −0.68683932191139646012807995310, −0.14988419013853458882722073454, 0.14988419013853458882722073454, 0.68683932191139646012807995310, 0.74143931724625328950622531049, 1.17484247657792286619657792801, 1.39861911881030270543555527029, 1.68999589125439346804391166113, 1.73178095800492796266937119658, 2.05810892733831576644429623908, 2.11258337538148573950512432305, 2.19390987259617518107973245289, 2.38375473016090557588198345211, 2.42212037504017540819117806438, 2.69193728838298276799641808520, 3.04089691281666289899115726355, 3.23826428875752414995723576962, 3.40677883811771216504954886130, 3.44400305551050215064272721348, 3.71646914728754141322944563918, 3.91568811073459788878805449103, 3.92340205981661260774166535316, 3.97299059955552216557339652913, 4.11100859110189924859848989142, 4.24816952159144144917680220250, 4.40989406108100661560738473135, 4.41860805657770448265467793288

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

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