Properties

Label 12-3332e6-1.1-c0e6-0-0
Degree $12$
Conductor $1.368\times 10^{21}$
Sign $1$
Analytic cond. $21.1432$
Root an. cond. $1.28952$
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 − 5·3-s + 5·6-s + 7-s + 15·9-s − 5·11-s + 5·13-s − 14-s − 17-s − 15·18-s − 5·21-s + 5·22-s + 2·23-s − 25-s − 5·26-s − 35·27-s + 2·31-s + 25·33-s + 34-s − 25·39-s + 5·42-s − 2·46-s + 50-s + 5·51-s − 2·53-s + 35·54-s − 2·62-s + ⋯
L(s)  = 1  − 2-s − 5·3-s + 5·6-s + 7-s + 15·9-s − 5·11-s + 5·13-s − 14-s − 17-s − 15·18-s − 5·21-s + 5·22-s + 2·23-s − 25-s − 5·26-s − 35·27-s + 2·31-s + 25·33-s + 34-s − 25·39-s + 5·42-s − 2·46-s + 50-s + 5·51-s − 2·53-s + 35·54-s − 2·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 7^{12} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(21.1432\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3332} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 7^{12} \cdot 17^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0003348519134\)
\(L(\frac12)\) \(\approx\) \(0.0003348519134\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
7 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
17 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
good3 \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
11 \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \)
13 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
67 \( ( 1 - T )^{6}( 1 + T )^{6} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
97 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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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

−4.92891450562438377191137973868, −4.54704064774992225096683047210, −4.40656388852732673871962817782, −4.37836332818426275165427131530, −4.37667816082238214350833080187, −4.01952632883530179143597315265, −4.01935910638189237534916565990, −3.70971037076037997130686393964, −3.55658383884694884193878397880, −3.31969567680309449118775789301, −3.29066087686150034731770777356, −3.03205587319924430838075138902, −2.95024773083053095708513365951, −2.75537476225048687331908415653, −2.11087475669093453480585777922, −2.10306961756013804235488997560, −2.05500255540446985087323566083, −1.98518704187982317773688742129, −1.79600152205025435696697965688, −1.17423292886629159964779486587, −1.15505477340255164107469844123, −1.06748305101418593736976033204, −0.936295276039083458896097147773, −0.74815926097795494371333911863, −0.01584294294396166476534273163, 0.01584294294396166476534273163, 0.74815926097795494371333911863, 0.936295276039083458896097147773, 1.06748305101418593736976033204, 1.15505477340255164107469844123, 1.17423292886629159964779486587, 1.79600152205025435696697965688, 1.98518704187982317773688742129, 2.05500255540446985087323566083, 2.10306961756013804235488997560, 2.11087475669093453480585777922, 2.75537476225048687331908415653, 2.95024773083053095708513365951, 3.03205587319924430838075138902, 3.29066087686150034731770777356, 3.31969567680309449118775789301, 3.55658383884694884193878397880, 3.70971037076037997130686393964, 4.01935910638189237534916565990, 4.01952632883530179143597315265, 4.37667816082238214350833080187, 4.37836332818426275165427131530, 4.40656388852732673871962817782, 4.54704064774992225096683047210, 4.92891450562438377191137973868

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

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