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Properties

Label 12-299e6-1.1-c0e6-0-0
Degree $12$
Conductor $7.145\times 10^{14}$
Sign $1$
Analytic cond. $1.10400\times 10^{-5}$
Root an. cond. $0.386290$
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  − 2·8-s − 3·23-s + 6·25-s − 2·27-s − 3·49-s − 6·59-s + 64-s + 3·101-s − 3·121-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 + 6·184-s + 191-s + 193-s + 197-s + 199-s − 12·200-s + ⋯
L(s)  = 1  − 2·8-s − 3·23-s + 6·25-s − 2·27-s − 3·49-s − 6·59-s + 64-s + 3·101-s − 3·121-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 + 6·184-s + 191-s + 193-s + 197-s + 199-s − 12·200-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(13^{6} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(1.10400\times 10^{-5}\)
Root analytic conductor: \(0.386290\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 13^{6} \cdot 23^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1816520305\)
\(L(\frac12)\) \(\approx\) \(0.1816520305\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + T^{3} + T^{6} \)
23 \( ( 1 + T + T^{2} )^{3} \)
good2 \( ( 1 + T^{3} + T^{6} )^{2} \)
3 \( ( 1 + T^{3} + T^{6} )^{2} \)
5 \( ( 1 - T )^{6}( 1 + T )^{6} \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
29 \( ( 1 + T^{3} + T^{6} )^{2} \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
41 \( ( 1 + T^{3} + T^{6} )^{2} \)
43 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
47 \( ( 1 + T^{3} + T^{6} )^{2} \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 + T + T^{2} )^{6} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
71 \( ( 1 + T^{3} + T^{6} )^{2} \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
97 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
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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

−6.59293363088018880145936642975, −6.49320073020378711582648325897, −6.34488300541015691053761942802, −6.26841501003280740617300131626, −6.14418389111510558898225052194, −5.65169547721349844421722682701, −5.64155818116268773100222929930, −5.37026584722359107940202369437, −5.33486246122980898888786396052, −4.98436037285445430790222961307, −4.73494891595770197915854247051, −4.50330399171643184386336081349, −4.49936446396339891242746601499, −4.19327593078679812009077170390, −4.11741859963816061257159406817, −3.39632345066396593428504732765, −3.36167765161284103148043506176, −3.19736159659980921420624301694, −3.11446746697763329318899357222, −3.03023510607917327183621616453, −2.50025538278488658699854800608, −2.32528372464057101963232958034, −1.86496868466883975423575023188, −1.59704747585539846253032562716, −1.23281169933623035559214643207, 1.23281169933623035559214643207, 1.59704747585539846253032562716, 1.86496868466883975423575023188, 2.32528372464057101963232958034, 2.50025538278488658699854800608, 3.03023510607917327183621616453, 3.11446746697763329318899357222, 3.19736159659980921420624301694, 3.36167765161284103148043506176, 3.39632345066396593428504732765, 4.11741859963816061257159406817, 4.19327593078679812009077170390, 4.49936446396339891242746601499, 4.50330399171643184386336081349, 4.73494891595770197915854247051, 4.98436037285445430790222961307, 5.33486246122980898888786396052, 5.37026584722359107940202369437, 5.64155818116268773100222929930, 5.65169547721349844421722682701, 6.14418389111510558898225052194, 6.26841501003280740617300131626, 6.34488300541015691053761942802, 6.49320073020378711582648325897, 6.59293363088018880145936642975

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

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