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Properties

Label 12-116e6-1.1-c0e6-0-0
Degree $12$
Conductor $2.436\times 10^{12}$
Sign $1$
Analytic cond. $3.76435\times 10^{-8}$
Root an. cond. $0.240606$
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-s − 2·5-s − 9-s + 2·10-s − 2·13-s − 2·17-s + 18-s + 25-s + 2·26-s − 29-s + 2·34-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 50-s + 5·53-s + 58-s − 2·61-s + 4·65-s + 5·73-s + 2·74-s + 2·82-s + 4·85-s − 2·89-s − 2·90-s + 5·97-s + ⋯
L(s)  = 1  − 2-s − 2·5-s − 9-s + 2·10-s − 2·13-s − 2·17-s + 18-s + 25-s + 2·26-s − 29-s + 2·34-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 50-s + 5·53-s + 58-s − 2·61-s + 4·65-s + 5·73-s + 2·74-s + 2·82-s + 4·85-s − 2·89-s − 2·90-s + 5·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 29^{6}\)
Sign: $1$
Analytic conductor: \(3.76435\times 10^{-8}\)
Root analytic conductor: \(0.240606\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 29^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01895423279\)
\(L(\frac12)\) \(\approx\) \(0.01895423279\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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

−8.262113482893172886022719944456, −7.61735796559339943625723157582, −7.52818446314234848157134324994, −7.30024813219237327687015242778, −7.22375586503258266377073054359, −7.16357847299187705940182612447, −6.89243826915145120674413000352, −6.61775284997599411631168994524, −6.29445753897241092991800488921, −6.13947680312980219114799289374, −5.80485425628311094218152912818, −5.66283865437418854723733575503, −5.07114890143137279357542216185, −4.99363779169325571996143279210, −4.98198964292263886909688124692, −4.81320037843895081907449787425, −4.25834907118962172763883245254, −4.05244636358745114329875164559, −3.78211432159173617962086038358, −3.57233705586428297510806529680, −3.36832508695949639784764112068, −2.97731371320272861811544350296, −2.31827131435495899951581404568, −2.26607837765745910984800191378, −1.98953106651989121279289648790, 1.98953106651989121279289648790, 2.26607837765745910984800191378, 2.31827131435495899951581404568, 2.97731371320272861811544350296, 3.36832508695949639784764112068, 3.57233705586428297510806529680, 3.78211432159173617962086038358, 4.05244636358745114329875164559, 4.25834907118962172763883245254, 4.81320037843895081907449787425, 4.98198964292263886909688124692, 4.99363779169325571996143279210, 5.07114890143137279357542216185, 5.66283865437418854723733575503, 5.80485425628311094218152912818, 6.13947680312980219114799289374, 6.29445753897241092991800488921, 6.61775284997599411631168994524, 6.89243826915145120674413000352, 7.16357847299187705940182612447, 7.22375586503258266377073054359, 7.30024813219237327687015242778, 7.52818446314234848157134324994, 7.61735796559339943625723157582, 8.262113482893172886022719944456

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

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