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Properties

Label 12-3267e6-1.1-c0e6-0-0
Degree $12$
Conductor $1.216\times 10^{21}$
Sign $1$
Analytic cond. $18.7861$
Root an. cond. $1.27688$
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  − 3·5-s + 3·25-s + 27-s + 3·31-s + 3·47-s + 3·59-s − 64-s − 6·67-s − 9·89-s + 6·97-s − 3·103-s − 6·113-s + 3·125-s + 127-s + 131-s − 3·135-s + 137-s + 139-s + 149-s + 151-s − 9·155-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 3·5-s + 3·25-s + 27-s + 3·31-s + 3·47-s + 3·59-s − 64-s − 6·67-s − 9·89-s + 6·97-s − 3·103-s − 6·113-s + 3·125-s + 127-s + 131-s − 3·135-s + 137-s + 139-s + 149-s + 151-s − 9·155-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(18.7861\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 11^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3706336337\)
\(L(\frac12)\) \(\approx\) \(0.3706336337\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{3} + T^{6} \)
11 \( 1 \)
good2 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
5 \( ( 1 + T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
7 \( 1 - T^{6} + T^{12} \)
13 \( 1 - T^{6} + T^{12} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 - T^{2} + T^{4} )^{3} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
37 \( ( 1 - T^{3} + T^{6} )^{2} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( 1 - T^{6} + T^{12} \)
47 \( ( 1 - T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
61 \( 1 - T^{6} + T^{12} \)
67 \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 - T^{2} + T^{4} )^{3} \)
79 \( 1 - T^{6} + T^{12} \)
83 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
89 \( ( 1 + T )^{6}( 1 + T + T^{2} )^{3} \)
97 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
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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

−4.54345887630728799729863111730, −4.54320704668175392370062628525, −4.27736987807645307424078805091, −4.08162600668159128520671153046, −4.06068017359096395147018676510, −3.98175038357704093492069605889, −3.90952031118968737506283365658, −3.90255912381001307393038261701, −3.65019555839717160049966558737, −3.20059540189718734433738952548, −3.12769073541937506830247325078, −3.04405335082360835359014651983, −2.97473811417611200024454703847, −2.78007155965612752466602809950, −2.61979814579062458905152562597, −2.52533307895191921441656512327, −2.34089851591561749872558152667, −2.06246410010373805515962438385, −1.88656625822407242026446373898, −1.31079925555514298866661564220, −1.29875893146418546164645615957, −1.25772394459459126218881973657, −1.19058572886936486759117684068, −0.57480218064960379523749741278, −0.28202531224940203087612249285, 0.28202531224940203087612249285, 0.57480218064960379523749741278, 1.19058572886936486759117684068, 1.25772394459459126218881973657, 1.29875893146418546164645615957, 1.31079925555514298866661564220, 1.88656625822407242026446373898, 2.06246410010373805515962438385, 2.34089851591561749872558152667, 2.52533307895191921441656512327, 2.61979814579062458905152562597, 2.78007155965612752466602809950, 2.97473811417611200024454703847, 3.04405335082360835359014651983, 3.12769073541937506830247325078, 3.20059540189718734433738952548, 3.65019555839717160049966558737, 3.90255912381001307393038261701, 3.90952031118968737506283365658, 3.98175038357704093492069605889, 4.06068017359096395147018676510, 4.08162600668159128520671153046, 4.27736987807645307424078805091, 4.54320704668175392370062628525, 4.54345887630728799729863111730

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

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