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Properties

Label 12-3332e6-1.1-c0e6-0-1
Degree $12$
Conductor $1.368\times 10^{21}$
Sign $1$
Analytic cond. $21.1432$
Root an. cond. $1.28952$
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-s + 5·3-s − 5·6-s − 7-s + 15·9-s + 5·11-s + 5·13-s + 14-s − 17-s − 15·18-s − 5·21-s − 5·22-s − 2·23-s − 25-s − 5·26-s + 35·27-s − 2·31-s + 25·33-s + 34-s + 25·39-s + 5·42-s + 2·46-s + 50-s − 5·51-s − 2·53-s − 35·54-s + 2·62-s + ⋯
L(s)  = 1  − 2-s + 5·3-s − 5·6-s − 7-s + 15·9-s + 5·11-s + 5·13-s + 14-s − 17-s − 15·18-s − 5·21-s − 5·22-s − 2·23-s − 25-s − 5·26-s + 35·27-s − 2·31-s + 25·33-s + 34-s + 25·39-s + 5·42-s + 2·46-s + 50-s − 5·51-s − 2·53-s − 35·54-s + 2·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 7^{12} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(21.1432\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 7^{12} \cdot 17^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(18.05972003\)
\(L(\frac12)\) \(\approx\) \(18.05972003\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
17 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
good3 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
11 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
13 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
23 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T )^{6}( 1 + T )^{6} \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
97 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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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

−4.29502192060727304995159696964, −4.28736852378233598901268534655, −4.25594542847401194036956686213, −4.04689507810119786006619538020, −3.99028868104225092765684075007, −3.91918708090996316887330666189, −3.73745350884794681555926611202, −3.59836164524781691976066744660, −3.41001227877780259508319474166, −3.30978354497273307790501486054, −3.25706194755196163332829296173, −3.16864521892414047524088377917, −3.16384483852969260352581428576, −2.57563752139687124372397925501, −2.54755603560408072253024734036, −2.54178433631714906095271459887, −1.92063486695949818513799242389, −1.82250609616824249987916364299, −1.81553218313765028260279962571, −1.78167761231273215605079634324, −1.56155869353784588912339738545, −1.29223192304533435226780931431, −1.27612303701564556537178517596, −1.10940514060402664446714313459, −0.906870331352341292476389969740, 0.906870331352341292476389969740, 1.10940514060402664446714313459, 1.27612303701564556537178517596, 1.29223192304533435226780931431, 1.56155869353784588912339738545, 1.78167761231273215605079634324, 1.81553218313765028260279962571, 1.82250609616824249987916364299, 1.92063486695949818513799242389, 2.54178433631714906095271459887, 2.54755603560408072253024734036, 2.57563752139687124372397925501, 3.16384483852969260352581428576, 3.16864521892414047524088377917, 3.25706194755196163332829296173, 3.30978354497273307790501486054, 3.41001227877780259508319474166, 3.59836164524781691976066744660, 3.73745350884794681555926611202, 3.91918708090996316887330666189, 3.99028868104225092765684075007, 4.04689507810119786006619538020, 4.25594542847401194036956686213, 4.28736852378233598901268534655, 4.29502192060727304995159696964

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

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