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Properties

Label 12-2523e6-1.1-c0e6-0-0
Degree $12$
Conductor $2.579\times 10^{20}$
Sign $1$
Analytic cond. $3.98516$
Root an. cond. $1.12211$
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  − 2-s + 3-s + 4-s − 6-s + 7-s − 11-s + 12-s + 13-s − 14-s + 6·17-s + 21-s + 22-s − 25-s − 26-s + 28-s − 33-s − 6·34-s + 39-s − 12·41-s − 42-s − 44-s − 47-s + 49-s + 50-s + 6·51-s + 52-s + 66-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 11-s + 12-s + 13-s − 14-s + 6·17-s + 21-s + 22-s − 25-s − 26-s + 28-s − 33-s − 6·34-s + 39-s − 12·41-s − 42-s − 44-s − 47-s + 49-s + 50-s + 6·51-s + 52-s + 66-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(3.98516\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 29^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.451721096\)
\(L(\frac12)\) \(\approx\) \(1.451721096\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
29 \( 1 \)
good2 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
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 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
11 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
13 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
17 \( ( 1 - T + T^{2} )^{6} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
41 \( ( 1 + T )^{12} \)
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 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
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 \( ( 1 - T )^{6}( 1 + T )^{6} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
67 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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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

−4.83393174029117698606661653496, −4.83062721460604297012433272404, −4.76503272919922634093086083256, −4.54058348087856805188807201916, −3.99952375747715530141429941475, −3.82716054622805759268503621344, −3.80782940858434823116273448153, −3.71849081307201234511565816427, −3.50239047565153207995928018258, −3.48980652330490030812847950301, −3.19619537241012289496920419142, −3.11349121948867926026910832906, −3.10479174910509789356280881722, −3.09780613640425390563840335533, −2.66898881811801412377626916242, −2.45922063554629742235526816478, −2.12464774612047355958199990126, −2.09428098455190613825654761652, −1.78415266742459309156314613085, −1.50462094085126324272121986629, −1.50422041062164268881109885976, −1.46367105230593812241287451839, −1.25384111235549620804038858402, −1.11613387356117617558569275510, −0.39937731199647628735268385611, 0.39937731199647628735268385611, 1.11613387356117617558569275510, 1.25384111235549620804038858402, 1.46367105230593812241287451839, 1.50422041062164268881109885976, 1.50462094085126324272121986629, 1.78415266742459309156314613085, 2.09428098455190613825654761652, 2.12464774612047355958199990126, 2.45922063554629742235526816478, 2.66898881811801412377626916242, 3.09780613640425390563840335533, 3.10479174910509789356280881722, 3.11349121948867926026910832906, 3.19619537241012289496920419142, 3.48980652330490030812847950301, 3.50239047565153207995928018258, 3.71849081307201234511565816427, 3.80782940858434823116273448153, 3.82716054622805759268503621344, 3.99952375747715530141429941475, 4.54058348087856805188807201916, 4.76503272919922634093086083256, 4.83062721460604297012433272404, 4.83393174029117698606661653496

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

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