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Properties

Label 12-7e18-1.1-c0e6-0-0
Degree $12$
Conductor $1.628\times 10^{15}$
Sign $1$
Analytic cond. $2.51598\times 10^{-5}$
Root an. cond. $0.413738$
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 + 4-s − 3·9-s + 11-s − 3·18-s + 22-s + 23-s − 3·25-s − 2·29-s − 3·36-s + 37-s − 2·43-s + 44-s + 46-s − 3·50-s + 53-s − 2·58-s + 67-s − 2·71-s + 74-s + 79-s + 3·81-s − 2·86-s + 92-s − 3·99-s − 3·100-s + 106-s + ⋯
L(s)  = 1  + 2-s + 4-s − 3·9-s + 11-s − 3·18-s + 22-s + 23-s − 3·25-s − 2·29-s − 3·36-s + 37-s − 2·43-s + 44-s + 46-s − 3·50-s + 53-s − 2·58-s + 67-s − 2·71-s + 74-s + 79-s + 3·81-s − 2·86-s + 92-s − 3·99-s − 3·100-s + 106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{18}\)
Sign: $1$
Analytic conductor: \(2.51598\times 10^{-5}\)
Root analytic conductor: \(0.413738\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{18} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3678683163\)
\(L(\frac12)\) \(\approx\) \(0.3678683163\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
3 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
5 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
13 \( ( 1 - T )^{6}( 1 + T )^{6} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
23 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
31 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
37 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
41 \( ( 1 - T )^{6}( 1 + T )^{6} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
47 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
53 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
73 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
79 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
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

−6.45183718027214081028098559589, −6.29187885231727753763823597668, −6.14160173287964578936094392757, −5.81834351031608906882421092064, −5.78624312852348378221258953801, −5.73213617047066158555142205173, −5.45699753797833725911274719724, −5.39656719309337282574786319919, −5.10832186684189240103502552380, −5.10723097923706954520388139442, −4.64579142667226724826640170439, −4.23925265064179308049453766669, −4.20409262435891659066249867286, −4.20303627284666249622610329499, −3.77550835514540896346174613858, −3.57963858302676031684651185298, −3.28654117561486097271160266144, −3.19361753404816666736204016897, −2.96022745269868503326464359224, −2.95717764115221183541719927057, −2.33283422461467011839925225113, −2.30709613013944813478349193405, −1.88557120157370809454424333276, −1.82895522013238947016832058001, −1.15881346181470270571965129211, 1.15881346181470270571965129211, 1.82895522013238947016832058001, 1.88557120157370809454424333276, 2.30709613013944813478349193405, 2.33283422461467011839925225113, 2.95717764115221183541719927057, 2.96022745269868503326464359224, 3.19361753404816666736204016897, 3.28654117561486097271160266144, 3.57963858302676031684651185298, 3.77550835514540896346174613858, 4.20303627284666249622610329499, 4.20409262435891659066249867286, 4.23925265064179308049453766669, 4.64579142667226724826640170439, 5.10723097923706954520388139442, 5.10832186684189240103502552380, 5.39656719309337282574786319919, 5.45699753797833725911274719724, 5.73213617047066158555142205173, 5.78624312852348378221258953801, 5.81834351031608906882421092064, 6.14160173287964578936094392757, 6.29187885231727753763823597668, 6.45183718027214081028098559589

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

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