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Properties

Label 12-327e6-1.1-c0e6-0-0
Degree $12$
Conductor $1.223\times 10^{15}$
Sign $1$
Analytic cond. $1.88898\times 10^{-5}$
Root an. cond. $0.403973$
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  − 3·4-s − 3·7-s + 3·16-s − 27-s + 9·28-s + 6·37-s + 3·49-s − 3·61-s + 2·64-s − 3·73-s − 3·79-s + 6·103-s + 3·108-s − 3·109-s − 9·112-s + 127-s + 131-s + 137-s + 139-s − 18·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + ⋯
L(s)  = 1  − 3·4-s − 3·7-s + 3·16-s − 27-s + 9·28-s + 6·37-s + 3·49-s − 3·61-s + 2·64-s − 3·73-s − 3·79-s + 6·103-s + 3·108-s − 3·109-s − 9·112-s + 127-s + 131-s + 137-s + 139-s − 18·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 109^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 109^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 109^{6}\)
Sign: $1$
Analytic conductor: \(1.88898\times 10^{-5}\)
Root analytic conductor: \(0.403973\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 109^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06155479389\)
\(L(\frac12)\) \(\approx\) \(0.06155479389\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T^{3} + T^{6} \)
109 \( ( 1 + T + T^{2} )^{3} \)
good2 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
13 \( ( 1 + T^{3} + T^{6} )^{2} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
41 \( ( 1 - T )^{6}( 1 + T )^{6} \)
43 \( ( 1 + T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 + T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
89 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
97 \( ( 1 + T^{3} + T^{6} )^{2} \)
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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

−6.44828649738504298749446915603, −6.23952794491228645706255650036, −6.11996400267261507175243328569, −6.05064956652600097344080768250, −6.03241619052378346314198635969, −5.77321768788320308870575220458, −5.53230330619079661081075372132, −5.33487710470243967833240595696, −4.77093294466043087626762956213, −4.76716037741080453583880091230, −4.65084575587586381209703119863, −4.63583662935218434366124114474, −4.32590652098531888466176723506, −4.01166352137983836379891895235, −3.96532377757361930321490868810, −3.89248678643693154304400326419, −3.40615112725190354561483430512, −3.24134701292766226999071646262, −2.98802182344657154975599558015, −2.89992595898240631270928883165, −2.57228649946871964806010764905, −2.35716625675619049078277098420, −1.84871207474608301217041957899, −1.29399629026774561618631666287, −0.64210657973540804225740003132, 0.64210657973540804225740003132, 1.29399629026774561618631666287, 1.84871207474608301217041957899, 2.35716625675619049078277098420, 2.57228649946871964806010764905, 2.89992595898240631270928883165, 2.98802182344657154975599558015, 3.24134701292766226999071646262, 3.40615112725190354561483430512, 3.89248678643693154304400326419, 3.96532377757361930321490868810, 4.01166352137983836379891895235, 4.32590652098531888466176723506, 4.63583662935218434366124114474, 4.65084575587586381209703119863, 4.76716037741080453583880091230, 4.77093294466043087626762956213, 5.33487710470243967833240595696, 5.53230330619079661081075372132, 5.77321768788320308870575220458, 6.03241619052378346314198635969, 6.05064956652600097344080768250, 6.11996400267261507175243328569, 6.23952794491228645706255650036, 6.44828649738504298749446915603

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

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