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Properties

Label 12-980e6-1.1-c0e6-0-0
Degree $12$
Conductor $8.858\times 10^{17}$
Sign $1$
Analytic cond. $0.0136867$
Root an. cond. $0.699345$
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-s − 5·6-s + 7-s + 15·9-s − 10-s + 14-s + 5·15-s + 15·18-s − 5·21-s + 2·23-s − 35·27-s + 5·29-s + 5·30-s − 35-s − 2·41-s − 5·42-s + 2·43-s − 15·45-s + 2·46-s + 2·47-s − 35·54-s + 5·58-s − 2·61-s + 15·63-s − 12·67-s + ⋯
L(s)  = 1  + 2-s − 5·3-s − 5-s − 5·6-s + 7-s + 15·9-s − 10-s + 14-s + 5·15-s + 15·18-s − 5·21-s + 2·23-s − 35·27-s + 5·29-s + 5·30-s − 35-s − 2·41-s − 5·42-s + 2·43-s − 15·45-s + 2·46-s + 2·47-s − 35·54-s + 5·58-s − 2·61-s + 15·63-s − 12·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 5^{6} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(0.0136867\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 5^{6} \cdot 7^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1862532662\)
\(L(\frac12)\) \(\approx\) \(0.1862532662\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
5 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
7 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
good3 \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
29 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
31 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
67 \( ( 1 + T )^{12} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
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

−5.36912787626619798347866556839, −5.33408840265871537983279562309, −5.16251400027676591725366808021, −4.94452096881994648187353747090, −4.86252003664822354211519184532, −4.76895444294960565116112356973, −4.70300686385153167233292855979, −4.46578934803074059405648017685, −4.41152545774057166764440618238, −4.18077976862656050492643992791, −4.15529939304290375261098484025, −4.12075175324556659238615922478, −3.88487061794276791787058824307, −3.32776932722988761998532540699, −3.07531136508925290234514039807, −3.06655090726072094378133430761, −2.97369842387203172125456815135, −2.64405627866272526515290270100, −2.06290134824308518935578865699, −1.99739946321435648735086389024, −1.36149034387707679416999370316, −1.34097608348028343016065956745, −1.30851887438826301727517610117, −1.12227874522549266783737319679, −0.50068067144684696359766649551, 0.50068067144684696359766649551, 1.12227874522549266783737319679, 1.30851887438826301727517610117, 1.34097608348028343016065956745, 1.36149034387707679416999370316, 1.99739946321435648735086389024, 2.06290134824308518935578865699, 2.64405627866272526515290270100, 2.97369842387203172125456815135, 3.06655090726072094378133430761, 3.07531136508925290234514039807, 3.32776932722988761998532540699, 3.88487061794276791787058824307, 4.12075175324556659238615922478, 4.15529939304290375261098484025, 4.18077976862656050492643992791, 4.41152545774057166764440618238, 4.46578934803074059405648017685, 4.70300686385153167233292855979, 4.76895444294960565116112356973, 4.86252003664822354211519184532, 4.94452096881994648187353747090, 5.16251400027676591725366808021, 5.33408840265871537983279562309, 5.36912787626619798347866556839

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

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