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Properties

Label 12-301e6-1.1-c0e6-0-0
Degree $12$
Conductor $7.437\times 10^{14}$
Sign $1$
Analytic cond. $1.14905\times 10^{-5}$
Root an. cond. $0.387580$
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  − 2·2-s + 4-s + 6·7-s − 9-s − 2·11-s − 12·14-s + 2·18-s + 4·22-s − 2·23-s − 25-s + 6·28-s − 2·29-s − 36-s − 2·37-s − 43-s − 2·44-s + 4·46-s + 21·49-s + 2·50-s + 5·53-s + 4·58-s − 6·63-s − 2·67-s − 2·71-s + 4·74-s − 12·77-s − 2·79-s + ⋯
L(s)  = 1  − 2·2-s + 4-s + 6·7-s − 9-s − 2·11-s − 12·14-s + 2·18-s + 4·22-s − 2·23-s − 25-s + 6·28-s − 2·29-s − 36-s − 2·37-s − 43-s − 2·44-s + 4·46-s + 21·49-s + 2·50-s + 5·53-s + 4·58-s − 6·63-s − 2·67-s − 2·71-s + 4·74-s − 12·77-s − 2·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{6} \cdot 43^{6}\)
Sign: $1$
Analytic conductor: \(1.14905\times 10^{-5}\)
Root analytic conductor: \(0.387580\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{6} \cdot 43^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08921687733\)
\(L(\frac12)\) \(\approx\) \(0.08921687733\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( ( 1 - T )^{6} \)
43 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
good2 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
3 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
11 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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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

−6.84096910382488117966024657397, −6.83344138977185905288248317510, −6.27862693468977544572725673501, −5.75593518421240812758718721869, −5.66906369772550549822150748873, −5.60601566093152097874722598855, −5.60158440120395343962078139220, −5.58481692280932537139618782314, −5.33329524829847081867501272660, −4.88862199295400287664079284874, −4.85015515502315111764356532018, −4.77696134238548258682130873157, −4.37086417744600458798373304721, −4.18669677668752026011384544754, −3.97310169624343039993261896349, −3.90817380802942986576832539027, −3.75428383471522006689923129145, −2.94450916017674938456775946091, −2.86618702816163580851389631141, −2.41678247331808358361943557180, −2.31618034094642495787376374267, −1.93229960416503960891380367380, −1.82423489981101654791309981305, −1.57660141722946161753596399855, −1.24837764466029275068300977539, 1.24837764466029275068300977539, 1.57660141722946161753596399855, 1.82423489981101654791309981305, 1.93229960416503960891380367380, 2.31618034094642495787376374267, 2.41678247331808358361943557180, 2.86618702816163580851389631141, 2.94450916017674938456775946091, 3.75428383471522006689923129145, 3.90817380802942986576832539027, 3.97310169624343039993261896349, 4.18669677668752026011384544754, 4.37086417744600458798373304721, 4.77696134238548258682130873157, 4.85015515502315111764356532018, 4.88862199295400287664079284874, 5.33329524829847081867501272660, 5.58481692280932537139618782314, 5.60158440120395343962078139220, 5.60601566093152097874722598855, 5.66906369772550549822150748873, 5.75593518421240812758718721869, 6.27862693468977544572725673501, 6.83344138977185905288248317510, 6.84096910382488117966024657397

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

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