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Properties

Label 12-2007e6-1.1-c0e6-0-0
Degree $12$
Conductor $6.536\times 10^{19}$
Sign $1$
Analytic cond. $1.00978$
Root an. cond. $1.00081$
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  + 4-s + 2·7-s − 2·19-s + 6·25-s + 2·28-s + 2·31-s − 2·37-s − 2·43-s + 49-s + 2·73-s − 2·76-s + 6·100-s − 2·109-s + 6·121-s + 2·124-s + 127-s + 131-s − 4·133-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + ⋯
L(s)  = 1  + 4-s + 2·7-s − 2·19-s + 6·25-s + 2·28-s + 2·31-s − 2·37-s − 2·43-s + 49-s + 2·73-s − 2·76-s + 6·100-s − 2·109-s + 6·121-s + 2·124-s + 127-s + 131-s − 4·133-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 223^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 223^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 223^{6}\)
Sign: $1$
Analytic conductor: \(1.00978\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 223^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.070426694\)
\(L(\frac12)\) \(\approx\) \(3.070426694\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( ( 1 + T )^{6} \)
good2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
5 \( ( 1 - T )^{6}( 1 + T )^{6} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
11 \( ( 1 - T )^{6}( 1 + T )^{6} \)
13 \( ( 1 - T )^{6}( 1 + T )^{6} \)
17 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
23 \( ( 1 - T )^{6}( 1 + T )^{6} \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
41 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T )^{6}( 1 + T )^{6} \)
61 \( ( 1 - T )^{6}( 1 + T )^{6} \)
67 \( ( 1 - T )^{6}( 1 + T )^{6} \)
71 \( ( 1 - T )^{6}( 1 + T )^{6} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
89 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
97 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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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

−5.01736579622662121151976988237, −4.73544192648831171775917684462, −4.67867208623404834743395622367, −4.64715203508313680488517615062, −4.40661518340663455128071464105, −4.32991469020337801119180432547, −4.30650958175196072601926702212, −4.03053973870126994144975349656, −3.56143545541767165109979559339, −3.49312396831364028628472554459, −3.48904375918196223123843472656, −3.23709019900996103047607764442, −2.95586799734051401962077380209, −2.92347842461185816926138548066, −2.90359826191420844740498937509, −2.52020698230579533138124979793, −2.32686831988777093178321621817, −2.20835984279170500546996557857, −1.87337273959216694144740636716, −1.86977031176743868347098900173, −1.83815540745740199642200787035, −1.26116329680899468838070771643, −1.19682122534145259993479432148, −0.985011821581742688426686727485, −0.76514182055565821395109245594, 0.76514182055565821395109245594, 0.985011821581742688426686727485, 1.19682122534145259993479432148, 1.26116329680899468838070771643, 1.83815540745740199642200787035, 1.86977031176743868347098900173, 1.87337273959216694144740636716, 2.20835984279170500546996557857, 2.32686831988777093178321621817, 2.52020698230579533138124979793, 2.90359826191420844740498937509, 2.92347842461185816926138548066, 2.95586799734051401962077380209, 3.23709019900996103047607764442, 3.48904375918196223123843472656, 3.49312396831364028628472554459, 3.56143545541767165109979559339, 4.03053973870126994144975349656, 4.30650958175196072601926702212, 4.32991469020337801119180432547, 4.40661518340663455128071464105, 4.64715203508313680488517615062, 4.67867208623404834743395622367, 4.73544192648831171775917684462, 5.01736579622662121151976988237

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

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