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Properties

Label 12-58e12-1.1-c0e6-0-3
Degree $12$
Conductor $1.449\times 10^{21}$
Sign $1$
Analytic cond. $22.3912$
Root an. cond. $1.29570$
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  + 2-s − 2·5-s − 9-s − 2·10-s − 2·13-s + 2·17-s − 18-s + 25-s − 2·26-s + 2·34-s + 2·37-s + 2·41-s + 2·45-s − 49-s + 50-s − 2·53-s + 2·61-s + 4·65-s + 2·73-s + 2·74-s + 2·82-s − 4·85-s + 2·89-s + 2·90-s + 2·97-s − 98-s + 2·101-s + ⋯
L(s)  = 1  + 2-s − 2·5-s − 9-s − 2·10-s − 2·13-s + 2·17-s − 18-s + 25-s − 2·26-s + 2·34-s + 2·37-s + 2·41-s + 2·45-s − 49-s + 50-s − 2·53-s + 2·61-s + 4·65-s + 2·73-s + 2·74-s + 2·82-s − 4·85-s + 2·89-s + 2·90-s + 2·97-s − 98-s + 2·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(22.3912\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 29^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.340601804\)
\(L(\frac12)\) \(\approx\) \(1.340601804\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
29 \( 1 \)
good3 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
5 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
59 \( ( 1 - T )^{6}( 1 + T )^{6} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
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

−4.71478186358330631940875829627, −4.66596518515900439457533423294, −4.36794616689495457125056161131, −4.24796427800841437476411372982, −3.92040373153952206219614471486, −3.91929941870457654757777852339, −3.85036768697026978985439394499, −3.62074761548328297505923954455, −3.55038957972861752794926742976, −3.47050412990326815575698737374, −3.42195977342186002909625246952, −2.89914791786947313753157940101, −2.88165185239810523497694604228, −2.62474725246308002747744619856, −2.59568170186257787890233213897, −2.56316009992434839693099110045, −2.54768953557826694665247258315, −2.05354899688969271206120151946, −1.86847447225533733364266645653, −1.59274431070614701842964239413, −1.52348831398832598186238308732, −1.23454065814373724682000790844, −0.78332839472093545930299812141, −0.66415324445354965764031549729, −0.46025518223602622248531292202, 0.46025518223602622248531292202, 0.66415324445354965764031549729, 0.78332839472093545930299812141, 1.23454065814373724682000790844, 1.52348831398832598186238308732, 1.59274431070614701842964239413, 1.86847447225533733364266645653, 2.05354899688969271206120151946, 2.54768953557826694665247258315, 2.56316009992434839693099110045, 2.59568170186257787890233213897, 2.62474725246308002747744619856, 2.88165185239810523497694604228, 2.89914791786947313753157940101, 3.42195977342186002909625246952, 3.47050412990326815575698737374, 3.55038957972861752794926742976, 3.62074761548328297505923954455, 3.85036768697026978985439394499, 3.91929941870457654757777852339, 3.92040373153952206219614471486, 4.24796427800841437476411372982, 4.36794616689495457125056161131, 4.66596518515900439457533423294, 4.71478186358330631940875829627

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

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