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Properties

Label 12-1944e6-1.1-c0e6-0-4
Degree $12$
Conductor $5.397\times 10^{19}$
Sign $1$
Analytic cond. $0.833912$
Root an. cond. $0.984978$
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  − 8-s + 6·11-s + 6·41-s − 3·43-s − 3·59-s − 3·67-s − 6·88-s + 3·89-s + 6·97-s + 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 8-s + 6·11-s + 6·41-s − 3·43-s − 3·59-s − 3·67-s − 6·88-s + 3·89-s + 6·97-s + 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{30}\)
Sign: $1$
Analytic conductor: \(0.833912\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{30} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.337640351\)
\(L(\frac12)\) \(\approx\) \(2.337640351\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.81770258966898918946966693137, −4.73040829549630836419242269738, −4.59686298497787781661486798272, −4.58815469625941458868910049291, −4.42809431644361422060776963832, −4.28360725707797285438794421648, −4.07270392229916274098671787366, −3.87622903704315979964898990198, −3.73347602510536179227797238339, −3.58947708986422348532754412386, −3.57093019607482095583095089548, −3.38187775229637607126614067051, −3.11273884170246690637738621808, −3.10424445265575968068013332402, −3.01224616975296691773175399169, −2.34076913402442625828128638181, −2.32717730817693094640086794488, −2.23806237755771189090842454127, −2.20895382781020691492426975177, −1.66721350363196016762576604878, −1.40741937195249149060067017829, −1.35981202647309825896168753469, −1.24723580579280442291750738281, −1.02700077572391596750237886066, −0.72109630646768855216018339324, 0.72109630646768855216018339324, 1.02700077572391596750237886066, 1.24723580579280442291750738281, 1.35981202647309825896168753469, 1.40741937195249149060067017829, 1.66721350363196016762576604878, 2.20895382781020691492426975177, 2.23806237755771189090842454127, 2.32717730817693094640086794488, 2.34076913402442625828128638181, 3.01224616975296691773175399169, 3.10424445265575968068013332402, 3.11273884170246690637738621808, 3.38187775229637607126614067051, 3.57093019607482095583095089548, 3.58947708986422348532754412386, 3.73347602510536179227797238339, 3.87622903704315979964898990198, 4.07270392229916274098671787366, 4.28360725707797285438794421648, 4.42809431644361422060776963832, 4.58815469625941458868910049291, 4.59686298497787781661486798272, 4.73040829549630836419242269738, 4.81770258966898918946966693137

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

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