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Properties

Label 12-1856e6-1.1-c0e6-0-0
Degree $12$
Conductor $4.088\times 10^{19}$
Sign $1$
Analytic cond. $0.631554$
Root an. cond. $0.962426$
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  + 2·5-s − 9-s + 2·13-s − 2·17-s + 25-s + 29-s + 2·37-s − 2·41-s − 2·45-s − 49-s − 5·53-s + 2·61-s + 4·65-s + 5·73-s − 4·85-s − 2·89-s + 5·97-s + 2·101-s − 5·109-s − 2·113-s − 2·117-s − 121-s + 127-s + 131-s + 137-s + 139-s + 2·145-s + ⋯
L(s)  = 1  + 2·5-s − 9-s + 2·13-s − 2·17-s + 25-s + 29-s + 2·37-s − 2·41-s − 2·45-s − 49-s − 5·53-s + 2·61-s + 4·65-s + 5·73-s − 4·85-s − 2·89-s + 5·97-s + 2·101-s − 5·109-s − 2·113-s − 2·117-s − 121-s + 127-s + 131-s + 137-s + 139-s + 2·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{36} \cdot 29^{6}\)
Sign: $1$
Analytic conductor: \(0.631554\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{36} \cdot 29^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.820915826\)
\(L(\frac12)\) \(\approx\) \(1.820915826\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
good3 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
41 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \)
59 \( ( 1 - T )^{6}( 1 + T )^{6} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
73 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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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

−5.10122894579329939120263907821, −4.94719996369427809411741744011, −4.66876623497616069309601969618, −4.58485122340742873241838176473, −4.58032189240322875801189173787, −4.25592842982005032787899473162, −4.20499878698334885868560809143, −4.02191864113879803354383958044, −3.77785340471309766806527586473, −3.48318108715966042457139637356, −3.37038309719593835125073094183, −3.31217466604819727718733129769, −3.30840804744627297131442995630, −2.83709722240483105810157874761, −2.79443863694213147961325159324, −2.69762829553008996335439194140, −2.28867404281208537476563313998, −2.16319791149616451417262272536, −2.06646359604052975073255466790, −1.79133123648937081140319502053, −1.78483012795885437338673264181, −1.57944896738247071962789934701, −1.11556240797236851068877923294, −1.00529995911701544077670993493, −0.58715733322287378105171032627, 0.58715733322287378105171032627, 1.00529995911701544077670993493, 1.11556240797236851068877923294, 1.57944896738247071962789934701, 1.78483012795885437338673264181, 1.79133123648937081140319502053, 2.06646359604052975073255466790, 2.16319791149616451417262272536, 2.28867404281208537476563313998, 2.69762829553008996335439194140, 2.79443863694213147961325159324, 2.83709722240483105810157874761, 3.30840804744627297131442995630, 3.31217466604819727718733129769, 3.37038309719593835125073094183, 3.48318108715966042457139637356, 3.77785340471309766806527586473, 4.02191864113879803354383958044, 4.20499878698334885868560809143, 4.25592842982005032787899473162, 4.58032189240322875801189173787, 4.58485122340742873241838176473, 4.66876623497616069309601969618, 4.94719996369427809411741744011, 5.10122894579329939120263907821

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

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