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Properties

Label 12-129e6-1.1-c0e6-0-0
Degree $12$
Conductor $4.608\times 10^{12}$
Sign $1$
Analytic cond. $7.12001\times 10^{-8}$
Root an. cond. $0.253730$
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-s − 4-s − 2·7-s + 12-s − 2·13-s − 2·19-s + 2·21-s − 25-s + 2·28-s + 5·31-s − 2·37-s + 2·39-s − 43-s + 49-s + 2·52-s + 2·57-s − 2·61-s − 2·67-s − 2·73-s + 75-s + 2·76-s − 2·79-s − 2·84-s + 4·91-s − 5·93-s − 2·97-s + 100-s + ⋯
L(s)  = 1  − 3-s − 4-s − 2·7-s + 12-s − 2·13-s − 2·19-s + 2·21-s − 25-s + 2·28-s + 5·31-s − 2·37-s + 2·39-s − 43-s + 49-s + 2·52-s + 2·57-s − 2·61-s − 2·67-s − 2·73-s + 75-s + 2·76-s − 2·79-s − 2·84-s + 4·91-s − 5·93-s − 2·97-s + 100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 43^{6}\)
Sign: $1$
Analytic conductor: \(7.12001\times 10^{-8}\)
Root analytic conductor: \(0.253730\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 43^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02916772711\)
\(L(\frac12)\) \(\approx\) \(0.02916772711\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
43 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
7 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
31 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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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

−7.86604371135424953251934593978, −7.61301906145116216345331273481, −7.07768067308750310483855913972, −7.00950574033900644621929818794, −6.96923331228537653412714555091, −6.91966399279496071088329594102, −6.52759954990724590546404907857, −6.20105180911059626308592776543, −6.05708096117740400824353938781, −6.05275396843483118535108134506, −5.78240941938146748824322621911, −5.49395294756135925311833737007, −5.33303418764230510184900226922, −4.80995547328830643512964708725, −4.54089190991418433684781540652, −4.49690869591238448603946969061, −4.49013472396186455404207866166, −4.33280687056394933560473687743, −3.78897603568362382235659278873, −3.30732714755598213119566821887, −3.00762260050070000034605042139, −2.88325771425224940394628159345, −2.86012454576471401070129813826, −1.97541040425816302260244792610, −1.86080065856534777136795850297, 1.86080065856534777136795850297, 1.97541040425816302260244792610, 2.86012454576471401070129813826, 2.88325771425224940394628159345, 3.00762260050070000034605042139, 3.30732714755598213119566821887, 3.78897603568362382235659278873, 4.33280687056394933560473687743, 4.49013472396186455404207866166, 4.49690869591238448603946969061, 4.54089190991418433684781540652, 4.80995547328830643512964708725, 5.33303418764230510184900226922, 5.49395294756135925311833737007, 5.78240941938146748824322621911, 6.05275396843483118535108134506, 6.05708096117740400824353938781, 6.20105180911059626308592776543, 6.52759954990724590546404907857, 6.91966399279496071088329594102, 6.96923331228537653412714555091, 7.00950574033900644621929818794, 7.07768067308750310483855913972, 7.61301906145116216345331273481, 7.86604371135424953251934593978

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

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