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Properties

Label 12-292e6-1.1-c0e6-0-0
Degree $12$
Conductor $6.199\times 10^{14}$
Sign $1$
Analytic cond. $9.57722\times 10^{-6}$
Root an. cond. $0.381742$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8-s − 3·9-s − 6·13-s − 3·29-s + 3·37-s + 3·41-s + 3·49-s + 3·61-s − 3·72-s + 3·81-s − 3·89-s + 3·101-s − 6·104-s − 6·113-s + 18·117-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 21·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 8-s − 3·9-s − 6·13-s − 3·29-s + 3·37-s + 3·41-s + 3·49-s + 3·61-s − 3·72-s + 3·81-s − 3·89-s + 3·101-s − 6·104-s − 6·113-s + 18·117-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 21·169-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 73^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 73^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 73^{6}\)
Sign: $1$
Analytic conductor: \(9.57722\times 10^{-6}\)
Root analytic conductor: \(0.381742\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 73^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1324762649\)
\(L(\frac12)\) \(\approx\) \(0.1324762649\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
73 \( 1 + T^{3} + T^{6} \)
good3 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T^{2} + T^{4} )^{3} \)
11 \( 1 - T^{6} + T^{12} \)
13 \( ( 1 + T )^{6}( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
19 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 + T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
31 \( 1 - T^{6} + T^{12} \)
37 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
41 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
43 \( ( 1 - T^{2} + T^{4} )^{3} \)
47 \( 1 - T^{6} + T^{12} \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 + T^{2} )^{6} \)
89 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
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

−6.96310598064322799082587662183, −6.70520627844820464004605730860, −6.27325586922871039671284459543, −6.07832907294508318411535398278, −5.65810656283610503419881226168, −5.64264061719470893608455850104, −5.56536580899237223629101767610, −5.54739064402416076731831119173, −5.45106549577479072523123944451, −4.93836181174907222355596841883, −4.85302705087345457570931739696, −4.74442522686034688852648680936, −4.23246623897311167497525927081, −4.18445349358480941188493682521, −4.12152951370281762632188025066, −4.08738912710648551545505118478, −3.30389782581599194371093683564, −3.15016745215710306689224161622, −2.82103044745311933512809568962, −2.72458474142751656274540105197, −2.48236510023002394201688031998, −2.32723443272388581723417847756, −2.10384253387558280108690209870, −2.03393529088393027297305313146, −0.888343645110811440146483830849, 0.888343645110811440146483830849, 2.03393529088393027297305313146, 2.10384253387558280108690209870, 2.32723443272388581723417847756, 2.48236510023002394201688031998, 2.72458474142751656274540105197, 2.82103044745311933512809568962, 3.15016745215710306689224161622, 3.30389782581599194371093683564, 4.08738912710648551545505118478, 4.12152951370281762632188025066, 4.18445349358480941188493682521, 4.23246623897311167497525927081, 4.74442522686034688852648680936, 4.85302705087345457570931739696, 4.93836181174907222355596841883, 5.45106549577479072523123944451, 5.54739064402416076731831119173, 5.56536580899237223629101767610, 5.64264061719470893608455850104, 5.65810656283610503419881226168, 6.07832907294508318411535398278, 6.27325586922871039671284459543, 6.70520627844820464004605730860, 6.96310598064322799082587662183

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

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