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Properties

Label 12-1944e6-1.1-c0e6-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·4-s + 2·8-s − 9·16-s + 3·29-s + 9·32-s − 9·58-s + 3·59-s + 3·64-s + 3·79-s + 3·103-s + 9·116-s − 9·118-s − 2·125-s + 127-s − 18·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 9·158-s + 163-s + 167-s − 3·169-s + 173-s + ⋯
L(s)  = 1  − 3·2-s + 3·4-s + 2·8-s − 9·16-s + 3·29-s + 9·32-s − 9·58-s + 3·59-s + 3·64-s + 3·79-s + 3·103-s + 9·116-s − 9·118-s − 2·125-s + 127-s − 18·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 9·158-s + 163-s + 167-s − 3·169-s + 173-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

\(L(\frac{1}{2})\) \(\approx\) \(0.2413348932\)
\(L(\frac12)\) \(\approx\) \(0.2413348932\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.90409837401965436412000179539, −4.86772839396710251696167306037, −4.84624954275607171154147555955, −4.59776392110822445229518739929, −4.51737409269969712745623413818, −4.10287485093965554084532949888, −4.07763783324926575746897106787, −3.95035057621037754001821971209, −3.84890155679467380457766656740, −3.70879025302866800840747896223, −3.48789550864974300351467525262, −3.31303211259440891380932879366, −2.92692556572061106053707352515, −2.76199603982275854906961217061, −2.65092140129290858932888959127, −2.60078891300650262781693739580, −2.28747096026134347296012203538, −1.97801602805078371994154140742, −1.78898556378285937405202770081, −1.74808370792006120066403392121, −1.61370256736324529589784915869, −1.05218041051058115435433916293, −0.944669311664349902682167997890, −0.74801962818237825558376684117, −0.65408932298723099128126462122, 0.65408932298723099128126462122, 0.74801962818237825558376684117, 0.944669311664349902682167997890, 1.05218041051058115435433916293, 1.61370256736324529589784915869, 1.74808370792006120066403392121, 1.78898556378285937405202770081, 1.97801602805078371994154140742, 2.28747096026134347296012203538, 2.60078891300650262781693739580, 2.65092140129290858932888959127, 2.76199603982275854906961217061, 2.92692556572061106053707352515, 3.31303211259440891380932879366, 3.48789550864974300351467525262, 3.70879025302866800840747896223, 3.84890155679467380457766656740, 3.95035057621037754001821971209, 4.07763783324926575746897106787, 4.10287485093965554084532949888, 4.51737409269969712745623413818, 4.59776392110822445229518739929, 4.84624954275607171154147555955, 4.86772839396710251696167306037, 4.90409837401965436412000179539

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

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