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Properties

Label 12-327e6-1.1-c0e6-0-1
Degree $12$
Conductor $1.223\times 10^{15}$
Sign $1$
Analytic cond. $1.88898\times 10^{-5}$
Root an. cond. $0.403973$
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  + 3·4-s − 3·7-s + 3·16-s + 27-s − 9·28-s − 6·37-s + 3·49-s − 3·61-s − 2·64-s + 3·73-s + 3·79-s − 6·103-s + 3·108-s + 3·109-s − 9·112-s + 127-s + 131-s + 137-s + 139-s − 18·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + ⋯
L(s)  = 1  + 3·4-s − 3·7-s + 3·16-s + 27-s − 9·28-s − 6·37-s + 3·49-s − 3·61-s − 2·64-s + 3·73-s + 3·79-s − 6·103-s + 3·108-s + 3·109-s − 9·112-s + 127-s + 131-s + 137-s + 139-s − 18·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 109^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 109^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 109^{6}\)
Sign: $1$
Analytic conductor: \(1.88898\times 10^{-5}\)
Root analytic conductor: \(0.403973\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 109^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3279543947\)
\(L(\frac12)\) \(\approx\) \(0.3279543947\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{3} + T^{6} \)
109 \( ( 1 - T + T^{2} )^{3} \)
good2 \( ( 1 - T^{2} + T^{4} )^{3} \)
5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
11 \( 1 - T^{6} + T^{12} \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{2} + T^{4} )^{3} \)
19 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
23 \( ( 1 - T^{2} + T^{4} )^{3} \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T^{3} + T^{6} )^{2} \)
37 \( ( 1 + T )^{6}( 1 + T^{3} + T^{6} ) \)
41 \( ( 1 + T^{2} )^{6} \)
43 \( ( 1 - T^{3} + T^{6} )^{2} \)
47 \( 1 - T^{6} + T^{12} \)
53 \( 1 - T^{6} + T^{12} \)
59 \( 1 - T^{6} + T^{12} \)
61 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
79 \( ( 1 - T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
89 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
97 \( ( 1 - T^{3} + T^{6} )^{2} \)
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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

−6.66026575661033693512733326576, −6.58192920312345757414773329301, −6.34093138807958092555677118087, −6.02599262683497503590154692843, −5.97498899279486872333287512101, −5.87616558965998739771999988023, −5.83703923098390678535868150765, −5.30024710020451418323314096438, −5.06377094358277328328366992908, −4.92589038139719394283417097099, −4.86451411761565733345167388443, −4.61559066775457815988716173797, −4.25789578904450902981116482302, −3.73942761173554667194033378443, −3.59945444173495950559078641696, −3.57452787459673205483191083685, −3.46113340227985790031771856771, −3.11064000397683426440119546371, −2.87233335650635994282891864306, −2.83639119490847620415274350698, −2.39839899821862538609512254240, −2.29533232849679845998752395850, −1.91367853609887871930860890472, −1.63660820470599085475125834462, −1.48796303032243066952913043664, 1.48796303032243066952913043664, 1.63660820470599085475125834462, 1.91367853609887871930860890472, 2.29533232849679845998752395850, 2.39839899821862538609512254240, 2.83639119490847620415274350698, 2.87233335650635994282891864306, 3.11064000397683426440119546371, 3.46113340227985790031771856771, 3.57452787459673205483191083685, 3.59945444173495950559078641696, 3.73942761173554667194033378443, 4.25789578904450902981116482302, 4.61559066775457815988716173797, 4.86451411761565733345167388443, 4.92589038139719394283417097099, 5.06377094358277328328366992908, 5.30024710020451418323314096438, 5.83703923098390678535868150765, 5.87616558965998739771999988023, 5.97498899279486872333287512101, 6.02599262683497503590154692843, 6.34093138807958092555677118087, 6.58192920312345757414773329301, 6.66026575661033693512733326576

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

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