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Properties

Label 12-259e6-1.1-c0e6-0-1
Degree $12$
Conductor $3.019\times 10^{14}$
Sign $1$
Analytic cond. $4.66381\times 10^{-6}$
Root an. cond. $0.359524$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3·4-s − 3·8-s − 12·16-s − 12·32-s − 6·37-s + 3·53-s + 5·64-s − 18·74-s − 3·79-s + 9·106-s − 3·107-s − 6·109-s + 127-s + 27·128-s + 131-s + 137-s + 139-s − 18·148-s + 149-s + 151-s + 157-s − 9·158-s + 163-s + 167-s + 173-s + 179-s + ⋯
L(s)  = 1  + 3·2-s + 3·4-s − 3·8-s − 12·16-s − 12·32-s − 6·37-s + 3·53-s + 5·64-s − 18·74-s − 3·79-s + 9·106-s − 3·107-s − 6·109-s + 127-s + 27·128-s + 131-s + 137-s + 139-s − 18·148-s + 149-s + 151-s + 157-s − 9·158-s + 163-s + 167-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{6} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(4.66381\times 10^{-6}\)
Root analytic conductor: \(0.359524\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{6} \cdot 37^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4298161915\)
\(L(\frac12)\) \(\approx\) \(0.4298161915\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−6.76004806702598065592279793921, −6.75472799354093078011665095774, −6.45270293842642255626497087706, −6.11747111035652999413829243545, −5.91707515570580336638314158990, −5.67366587395021158131894445466, −5.61439073517865027215635615667, −5.32616395134281936116805121071, −5.32496961865136083294676028521, −5.28663243884922333475609583068, −5.05518284087665211421807276658, −4.68438512988782542150364023476, −4.50777517946080358101744455201, −4.20522045084491248184233413522, −4.07122471888827544215301308467, −3.84758680117237124626302359585, −3.72426008546129251104533196805, −3.65298061487029830057269973265, −3.11677351118433584655623335316, −3.07426767971425674511930196191, −2.89162454209470909924750427662, −2.60387173103263702446257068115, −2.39289469795024541590199380293, −1.75166113096160994452500766015, −1.57721265420091151683694183684, 1.57721265420091151683694183684, 1.75166113096160994452500766015, 2.39289469795024541590199380293, 2.60387173103263702446257068115, 2.89162454209470909924750427662, 3.07426767971425674511930196191, 3.11677351118433584655623335316, 3.65298061487029830057269973265, 3.72426008546129251104533196805, 3.84758680117237124626302359585, 4.07122471888827544215301308467, 4.20522045084491248184233413522, 4.50777517946080358101744455201, 4.68438512988782542150364023476, 5.05518284087665211421807276658, 5.28663243884922333475609583068, 5.32496961865136083294676028521, 5.32616395134281936116805121071, 5.61439073517865027215635615667, 5.67366587395021158131894445466, 5.91707515570580336638314158990, 6.11747111035652999413829243545, 6.45270293842642255626497087706, 6.75472799354093078011665095774, 6.76004806702598065592279793921

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

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