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Properties

Label 12-3e36-1.1-c0e6-0-0
Degree $12$
Conductor $1.501\times 10^{17}$
Sign $1$
Analytic cond. $0.00231903$
Root an. cond. $0.603173$
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  + 3·19-s + 3·37-s − 64-s − 6·73-s − 6·109-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 3·19-s + 3·37-s − 64-s − 6·73-s − 6·109-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{36}\)
Sign: $1$
Analytic conductor: \(0.00231903\)
Root analytic conductor: \(0.603173\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{36} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7062221114\)
\(L(\frac12)\) \(\approx\) \(0.7062221114\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 + T^{3} + T^{6} )^{2} \)
11 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
13 \( ( 1 + T^{3} + T^{6} )^{2} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 + T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 + T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( ( 1 + T + T^{2} )^{6} \)
79 \( ( 1 + T^{3} + T^{6} )^{2} \)
83 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
89 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
97 \( ( 1 + T^{3} + T^{6} )^{2} \)
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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

−5.85019789089647864158040672938, −5.70568743806275732140333888411, −5.35652577258407168235496086842, −5.24195622390657075321901953825, −5.09415604745961991801379207341, −5.07687980756451806703374478907, −4.94697130196106096515591567895, −4.43165924689846831773873829699, −4.35590957197330934294767898556, −4.31598678520314317107371364768, −4.02644344698241976310623600268, −3.94674040639275022643603232554, −3.91224601865939168016083475387, −3.30006905457313040992511890378, −3.18051263007431464899492891317, −3.09113650625619699444370032892, −2.89788122762755178437030296667, −2.83779643984027513884424658112, −2.63873397899260259435274010216, −2.25700220714382018263247780752, −1.99295865601455482815840469047, −1.69316801652212957212058389030, −1.33480327883852614411324441832, −1.11039595924581354552842731452, −1.07006860452449239445642996379, 1.07006860452449239445642996379, 1.11039595924581354552842731452, 1.33480327883852614411324441832, 1.69316801652212957212058389030, 1.99295865601455482815840469047, 2.25700220714382018263247780752, 2.63873397899260259435274010216, 2.83779643984027513884424658112, 2.89788122762755178437030296667, 3.09113650625619699444370032892, 3.18051263007431464899492891317, 3.30006905457313040992511890378, 3.91224601865939168016083475387, 3.94674040639275022643603232554, 4.02644344698241976310623600268, 4.31598678520314317107371364768, 4.35590957197330934294767898556, 4.43165924689846831773873829699, 4.94697130196106096515591567895, 5.07687980756451806703374478907, 5.09415604745961991801379207341, 5.24195622390657075321901953825, 5.35652577258407168235496086842, 5.70568743806275732140333888411, 5.85019789089647864158040672938

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

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