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Properties

Label 12-1368e6-1.1-c0e6-0-10
Degree $12$
Conductor $6.554\times 10^{18}$
Sign $1$
Analytic cond. $0.101264$
Root an. cond. $0.826269$
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  + 6·3-s − 8-s + 21·9-s − 6·24-s + 56·27-s − 3·41-s − 3·49-s − 3·59-s − 3·67-s − 21·72-s − 3·73-s + 126·81-s − 3·97-s + 3·107-s − 18·123-s + 127-s + 131-s + 137-s + 139-s − 18·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 18·177-s + ⋯
L(s)  = 1  + 6·3-s − 8-s + 21·9-s − 6·24-s + 56·27-s − 3·41-s − 3·49-s − 3·59-s − 3·67-s − 21·72-s − 3·73-s + 126·81-s − 3·97-s + 3·107-s − 18·123-s + 127-s + 131-s + 137-s + 139-s − 18·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 18·177-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(0.101264\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{12} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(11.99079911\)
\(L(\frac12)\) \(\approx\) \(11.99079911\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{3} + T^{6} \)
3 \( ( 1 - T )^{6} \)
19 \( 1 + T^{3} + T^{6} \)
good5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 + T^{3} + T^{6} )^{2} \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T )^{6}( 1 + T )^{6} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
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 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 + T^{3} + T^{6} )^{2} \)
89 \( ( 1 + T^{3} + T^{6} )^{2} \)
97 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
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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

−5.04144125733701077584601740588, −5.03280134109724917289783525895, −4.75983082028672499142384832834, −4.67394843388091085531201171298, −4.53425820708742720647147618492, −4.28381027611245566600458924389, −4.15604159493434402295917559932, −4.13177413080031401308217824498, −3.85776214120065187443612338605, −3.58658584976803941919363871206, −3.49145132306645658100101960287, −3.38626113087263582932418115737, −3.22134556471724146144083999877, −3.07329726196318879561870020826, −3.06208002041103772966542100755, −2.89569972333064733556430226715, −2.48896422121217798282457803573, −2.48368565352088062021267999577, −2.40723046801462471112230810726, −2.18768351662667710663406881287, −1.69916276353853756118770485195, −1.51026972780413350786212692949, −1.43845981839743621590915414958, −1.43164168372506385630254904679, −1.37812690684402521232657985478, 1.37812690684402521232657985478, 1.43164168372506385630254904679, 1.43845981839743621590915414958, 1.51026972780413350786212692949, 1.69916276353853756118770485195, 2.18768351662667710663406881287, 2.40723046801462471112230810726, 2.48368565352088062021267999577, 2.48896422121217798282457803573, 2.89569972333064733556430226715, 3.06208002041103772966542100755, 3.07329726196318879561870020826, 3.22134556471724146144083999877, 3.38626113087263582932418115737, 3.49145132306645658100101960287, 3.58658584976803941919363871206, 3.85776214120065187443612338605, 4.13177413080031401308217824498, 4.15604159493434402295917559932, 4.28381027611245566600458924389, 4.53425820708742720647147618492, 4.67394843388091085531201171298, 4.75983082028672499142384832834, 5.03280134109724917289783525895, 5.04144125733701077584601740588

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

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