Properties

Label 12-912e6-1.1-c0e6-0-1
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $0.00889020$
Root an. cond. $0.674646$
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  − 3·13-s + 3·19-s − 27-s − 3·43-s − 6·61-s + 3·67-s + 3·73-s + 6·79-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 3·13-s + 3·19-s − 27-s − 3·43-s − 6·61-s + 3·67-s + 3·73-s + 6·79-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(0.00889020\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5718206323\)
\(L(\frac12)\) \(\approx\) \(0.5718206323\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T^{3} + T^{6} \)
19 \( ( 1 - T + T^{2} )^{3} \)
good5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 - T^{2} + T^{4} )^{3} \)
13 \( ( 1 + T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
23 \( 1 - T^{6} + T^{12} \)
29 \( 1 - T^{6} + T^{12} \)
31 \( ( 1 - T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
41 \( 1 - T^{6} + T^{12} \)
43 \( ( 1 + T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
47 \( 1 - T^{6} + T^{12} \)
53 \( 1 - T^{6} + T^{12} \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \)
67 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
79 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{2} + T^{4} )^{3} \)
89 \( 1 - T^{6} + T^{12} \)
97 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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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

−5.66900981762752182794564855466, −5.24014512089965221020245744504, −5.17166781731405271636363857033, −5.02693358736580162719837644662, −4.95677267093251883962064193656, −4.89644599135766081528685512061, −4.71136406018848628283843838057, −4.60422785778914165055794953683, −4.44810605505007429666808240124, −3.94567716652067600437599604543, −3.80858564929013575462270485963, −3.74216452616591693323322566386, −3.40817368674870713843355693494, −3.38393479511351351385290943884, −3.21362123084486307079913040451, −3.02827478829573137930144099920, −2.89034679860065617475391385705, −2.47292719545086264288687281225, −2.28345782723894083283013187983, −2.10856510065065962670930584359, −2.07073964914053771211321183476, −1.76660595903075830785642840647, −1.30744447492846902158167675822, −1.15741630604833331333751156162, −0.62677652499218229339179798872, 0.62677652499218229339179798872, 1.15741630604833331333751156162, 1.30744447492846902158167675822, 1.76660595903075830785642840647, 2.07073964914053771211321183476, 2.10856510065065962670930584359, 2.28345782723894083283013187983, 2.47292719545086264288687281225, 2.89034679860065617475391385705, 3.02827478829573137930144099920, 3.21362123084486307079913040451, 3.38393479511351351385290943884, 3.40817368674870713843355693494, 3.74216452616591693323322566386, 3.80858564929013575462270485963, 3.94567716652067600437599604543, 4.44810605505007429666808240124, 4.60422785778914165055794953683, 4.71136406018848628283843838057, 4.89644599135766081528685512061, 4.95677267093251883962064193656, 5.02693358736580162719837644662, 5.17166781731405271636363857033, 5.24014512089965221020245744504, 5.66900981762752182794564855466

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

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