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Properties

Label 12-387e6-1.1-c0e6-0-0
Degree $12$
Conductor $3.359\times 10^{15}$
Sign $1$
Analytic cond. $5.19049\times 10^{-5}$
Root an. cond. $0.439474$
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  − 4-s + 2·13-s − 25-s − 5·31-s + 43-s − 49-s − 2·52-s + 2·67-s + 2·79-s − 2·97-s + 100-s − 2·103-s + 5·109-s + 121-s + 5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 172-s + 173-s + ⋯
L(s)  = 1  − 4-s + 2·13-s − 25-s − 5·31-s + 43-s − 49-s − 2·52-s + 2·67-s + 2·79-s − 2·97-s + 100-s − 2·103-s + 5·109-s + 121-s + 5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 172-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 43^{6}\)
Sign: $1$
Analytic conductor: \(5.19049\times 10^{-5}\)
Root analytic conductor: \(0.439474\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 43^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2447973909\)
\(L(\frac12)\) \(\approx\) \(0.2447973909\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
17 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
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 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
31 \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
41 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
47 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
53 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
59 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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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

−6.52092312491202000515826251929, −6.04501949723655223305687385693, −5.89737048620578327341346174860, −5.85968276153256928277690616882, −5.76766154129259029203947750022, −5.58588154915079513999396614874, −5.44706235159767669221482363742, −5.06195962311433660364737582025, −4.83867006180048377015274289916, −4.80232048095111663789828324354, −4.68853446828249108260348600447, −4.28780999282025239580372966229, −4.13422779959580015541245015690, −3.78822688227793451436260835727, −3.72836273336520525067930765893, −3.57228237008269759645021535471, −3.54074184617160397923076315995, −3.23430110657402909461712658325, −3.03713585885310433316305981298, −2.39982453105031808256538599607, −2.26978326166163652338272170894, −2.00546506124317938350757554901, −1.90305191097641647955838317118, −1.31447089462645538408403416119, −1.13318946191385263806797014892, 1.13318946191385263806797014892, 1.31447089462645538408403416119, 1.90305191097641647955838317118, 2.00546506124317938350757554901, 2.26978326166163652338272170894, 2.39982453105031808256538599607, 3.03713585885310433316305981298, 3.23430110657402909461712658325, 3.54074184617160397923076315995, 3.57228237008269759645021535471, 3.72836273336520525067930765893, 3.78822688227793451436260835727, 4.13422779959580015541245015690, 4.28780999282025239580372966229, 4.68853446828249108260348600447, 4.80232048095111663789828324354, 4.83867006180048377015274289916, 5.06195962311433660364737582025, 5.44706235159767669221482363742, 5.58588154915079513999396614874, 5.76766154129259029203947750022, 5.85968276153256928277690616882, 5.89737048620578327341346174860, 6.04501949723655223305687385693, 6.52092312491202000515826251929

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

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