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Properties

Label 12-1368e6-1.1-c0e6-0-0
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  − 8-s − 27-s − 3·41-s − 3·49-s − 3·59-s − 3·67-s − 3·73-s − 3·97-s − 6·107-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 216-s + ⋯
L(s)  = 1  − 8-s − 27-s − 3·41-s − 3·49-s − 3·59-s − 3·67-s − 3·73-s − 3·97-s − 6·107-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 216-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

\(L(\frac{1}{2})\) \(\approx\) \(0.1435909897\)
\(L(\frac12)\) \(\approx\) \(0.1435909897\)
\(L(1)\) 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 \( 1 + T^{3} + T^{6} \)
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

−5.24428967078670367066662669490, −5.17913680831518238405860598478, −5.06567192777839128786242979895, −4.91672145825870559323984662226, −4.53601154462552804002952560531, −4.50057603988980234923567831222, −4.32842368214115029869908058915, −4.05474161767340251238105617188, −4.04202721541876520505888493464, −3.80071021914470413803482622418, −3.74526455105609671537830422025, −3.55567064592861768381024335324, −3.00728146216149728087787171911, −2.97927172552445241011113117541, −2.95612572687596962168003483353, −2.87097021543909185438652324061, −2.80071186756687473897408780898, −2.68605637015677730009040488767, −1.83582363627582254173468044454, −1.83561561708543029341117683936, −1.72341843891266410999098124536, −1.62327040332463009654822383384, −1.49456594073301148914440252572, −1.04753531556126707889247878159, −0.19785671053864246233162746748, 0.19785671053864246233162746748, 1.04753531556126707889247878159, 1.49456594073301148914440252572, 1.62327040332463009654822383384, 1.72341843891266410999098124536, 1.83561561708543029341117683936, 1.83582363627582254173468044454, 2.68605637015677730009040488767, 2.80071186756687473897408780898, 2.87097021543909185438652324061, 2.95612572687596962168003483353, 2.97927172552445241011113117541, 3.00728146216149728087787171911, 3.55567064592861768381024335324, 3.74526455105609671537830422025, 3.80071021914470413803482622418, 4.04202721541876520505888493464, 4.05474161767340251238105617188, 4.32842368214115029869908058915, 4.50057603988980234923567831222, 4.53601154462552804002952560531, 4.91672145825870559323984662226, 5.06567192777839128786242979895, 5.17913680831518238405860598478, 5.24428967078670367066662669490

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

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