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Properties

Label 12-207e6-1.1-c0e6-0-0
Degree $12$
Conductor $7.867\times 10^{13}$
Sign $1$
Analytic cond. $1.21552\times 10^{-6}$
Root an. cond. $0.321413$
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  − 2·8-s − 3·23-s − 3·25-s − 27-s − 3·49-s + 3·59-s + 64-s + 3·101-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 6·184-s + 191-s + 193-s + 197-s + 199-s + 6·200-s + ⋯
L(s)  = 1  − 2·8-s − 3·23-s − 3·25-s − 27-s − 3·49-s + 3·59-s + 64-s + 3·101-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 6·184-s + 191-s + 193-s + 197-s + 199-s + 6·200-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(1.21552\times 10^{-6}\)
Root analytic conductor: \(0.321413\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 23^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1042398809\)
\(L(\frac12)\) \(\approx\) \(0.1042398809\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T^{3} + T^{6} \)
23 \( ( 1 + T + T^{2} )^{3} \)
good2 \( ( 1 + T^{3} + T^{6} )^{2} \)
5 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 + T^{3} + T^{6} )^{2} \)
17 \( ( 1 - T )^{6}( 1 + T )^{6} \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
29 \( ( 1 + T^{3} + T^{6} )^{2} \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 + T^{3} + T^{6} )^{2} \)
43 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
47 \( ( 1 + T^{3} + T^{6} )^{2} \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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 \( ( 1 + T^{3} + T^{6} )^{2} \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
83 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
89 \( ( 1 - T )^{6}( 1 + T )^{6} \)
97 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
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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

−7.19170087334879226206411709511, −6.75111261241441636550738520954, −6.63790554828572067453005974501, −6.50655848760131952298010046329, −6.37725458535952693750632654202, −6.21378919839779505524566768705, −5.83323076343614519974055294283, −5.80370916133098730207207173386, −5.64414085076253983273149391896, −5.61542004363749576648631745284, −5.25379888439900349615614016668, −4.95422883148007704837880648110, −4.70521179965447883351608023496, −4.47684233937124717347178729887, −4.20910329994878808624538866646, −3.87015542094150353159278466892, −3.67297728891292611189604572878, −3.61010769354837001529069428782, −3.59256274110444935456057849803, −2.93369459737726509413576750423, −2.83838046183866168281928179967, −2.31067133706731657237815546333, −2.25393059789021078712292873229, −1.73225099048033422953589967046, −1.71818216672499630298919976284, 1.71818216672499630298919976284, 1.73225099048033422953589967046, 2.25393059789021078712292873229, 2.31067133706731657237815546333, 2.83838046183866168281928179967, 2.93369459737726509413576750423, 3.59256274110444935456057849803, 3.61010769354837001529069428782, 3.67297728891292611189604572878, 3.87015542094150353159278466892, 4.20910329994878808624538866646, 4.47684233937124717347178729887, 4.70521179965447883351608023496, 4.95422883148007704837880648110, 5.25379888439900349615614016668, 5.61542004363749576648631745284, 5.64414085076253983273149391896, 5.80370916133098730207207173386, 5.83323076343614519974055294283, 6.21378919839779505524566768705, 6.37725458535952693750632654202, 6.50655848760131952298010046329, 6.63790554828572067453005974501, 6.75111261241441636550738520954, 7.19170087334879226206411709511

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

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