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Properties

Label 12-297e6-1.1-c0e6-0-0
Degree $12$
Conductor $6.863\times 10^{14}$
Sign $1$
Analytic cond. $1.06042\times 10^{-5}$
Root an. cond. $0.384996$
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  − 3·5-s + 3·25-s − 27-s − 3·31-s − 3·47-s − 3·59-s − 64-s + 6·67-s + 3·89-s + 6·97-s − 3·103-s + 6·113-s + 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 + 3·89-s + 6·97-s − 3·103-s + 6·113-s + 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

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

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(1.06042\times 10^{-5}\)
Root analytic conductor: \(0.384996\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 11^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1098942121\)
\(L(\frac12)\) \(\approx\) \(0.1098942121\)
\(L(1)\) 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 \( 1 + T^{3} + T^{6} \)
good2 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
5 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
23 \( ( 1 + T^{3} + T^{6} )^{2} \)
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 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
47 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 + T^{3} + T^{6} )^{2} \)
59 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 + T^{3} + T^{6} )^{2} \)
73 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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

−6.85423897090158580015044351358, −6.43120060138251007139008625511, −6.40881090642502171409726828797, −6.16493453451054474175619928489, −5.97799461313668822035095125698, −5.84471138623774864204276039501, −5.55918773180648514914797225710, −5.40461400541483976317311699937, −4.92723638224117684674520181114, −4.90346390341517136731147186619, −4.86139087454397805016920937725, −4.61173491729997113317236599889, −4.36598838527626367752588302057, −4.04047450003798566291325691540, −3.96739365909454404448995398838, −3.59681502528997047131938143207, −3.40459537878854720293399672875, −3.37820722857511416735701443745, −3.36417730849738865534218497816, −3.16794659852556745005572638295, −2.29619675943427701773647904468, −2.19370799218050459251125349540, −1.91061384467226185823412321544, −1.79216800080163308172253952172, −0.864307895157398535155276217590, 0.864307895157398535155276217590, 1.79216800080163308172253952172, 1.91061384467226185823412321544, 2.19370799218050459251125349540, 2.29619675943427701773647904468, 3.16794659852556745005572638295, 3.36417730849738865534218497816, 3.37820722857511416735701443745, 3.40459537878854720293399672875, 3.59681502528997047131938143207, 3.96739365909454404448995398838, 4.04047450003798566291325691540, 4.36598838527626367752588302057, 4.61173491729997113317236599889, 4.86139087454397805016920937725, 4.90346390341517136731147186619, 4.92723638224117684674520181114, 5.40461400541483976317311699937, 5.55918773180648514914797225710, 5.84471138623774864204276039501, 5.97799461313668822035095125698, 6.16493453451054474175619928489, 6.40881090642502171409726828797, 6.43120060138251007139008625511, 6.85423897090158580015044351358

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

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