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Properties

Label 12-58e12-1.1-c0e6-0-7
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 + 5·5-s − 9-s − 5·10-s + 5·13-s − 2·17-s + 18-s + 15·25-s − 5·26-s + 2·34-s + 5·37-s − 2·41-s − 5·45-s − 49-s − 15·50-s − 2·53-s + 5·61-s + 25·65-s − 2·73-s − 5·74-s + 2·82-s − 10·85-s + 5·89-s + 5·90-s − 2·97-s + 98-s + 5·101-s + ⋯
L(s)  = 1  − 2-s + 5·5-s − 9-s − 5·10-s + 5·13-s − 2·17-s + 18-s + 15·25-s − 5·26-s + 2·34-s + 5·37-s − 2·41-s − 5·45-s − 49-s − 15·50-s − 2·53-s + 5·61-s + 25·65-s − 2·73-s − 5·74-s + 2·82-s − 10·85-s + 5·89-s + 5·90-s − 2·97-s + 98-s + 5·101-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

\(L(\frac{1}{2})\) \(\approx\) \(5.396418244\)
\(L(\frac12)\) \(\approx\) \(5.396418244\)
\(L(1)\) 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 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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

−4.60127701350787551700215061896, −4.59633157817008055152885453517, −4.59523089867574250956592843294, −4.24351784164320701598000892300, −4.09201211357692883431500460796, −3.83157896382912182476790309270, −3.65478700234436113141541644284, −3.64260597683414204613203554132, −3.55776239475142403926186745685, −3.06011333904993834492851918909, −2.96894162155472768342237359063, −2.89844209916642136071620447425, −2.87492795964932702809552206742, −2.86562893321958250133473648733, −2.32279957545436224843053379650, −2.11256882603814111306834582596, −2.10560163405798845240666623471, −1.96867285347024772883967750196, −1.95517833113040824264617712515, −1.80030725281333404655264044284, −1.25811309088899590364020007235, −1.14399601749390103550408592571, −1.11549733468413786357160728889, −0.900949698853721951076882747316, −0.886895070456441456411712866099, 0.886895070456441456411712866099, 0.900949698853721951076882747316, 1.11549733468413786357160728889, 1.14399601749390103550408592571, 1.25811309088899590364020007235, 1.80030725281333404655264044284, 1.95517833113040824264617712515, 1.96867285347024772883967750196, 2.10560163405798845240666623471, 2.11256882603814111306834582596, 2.32279957545436224843053379650, 2.86562893321958250133473648733, 2.87492795964932702809552206742, 2.89844209916642136071620447425, 2.96894162155472768342237359063, 3.06011333904993834492851918909, 3.55776239475142403926186745685, 3.64260597683414204613203554132, 3.65478700234436113141541644284, 3.83157896382912182476790309270, 4.09201211357692883431500460796, 4.24351784164320701598000892300, 4.59523089867574250956592843294, 4.59633157817008055152885453517, 4.60127701350787551700215061896

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

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