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Properties

Label 12-299e6-1.1-c0e6-0-1
Degree $12$
Conductor $7.145\times 10^{14}$
Sign $1$
Analytic cond. $1.10400\times 10^{-5}$
Root an. cond. $0.386290$
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  − 3·23-s − 6·25-s + 2·27-s + 3·49-s − 64-s − 3·101-s + 3·121-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 + ⋯
L(s)  = 1  − 3·23-s − 6·25-s + 2·27-s + 3·49-s − 64-s − 3·101-s + 3·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(13^{6} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(1.10400\times 10^{-5}\)
Root analytic conductor: \(0.386290\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 13^{6} \cdot 23^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2004228060\)
\(L(\frac12)\) \(\approx\) \(0.2004228060\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + T^{3} + T^{6} \)
23 \( ( 1 + T + T^{2} )^{3} \)
good2 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
3 \( ( 1 - T^{3} + T^{6} )^{2} \)
5 \( ( 1 + T^{2} )^{6} \)
7 \( ( 1 - T^{2} + T^{4} )^{3} \)
11 \( ( 1 - T^{2} + T^{4} )^{3} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 - T^{2} + T^{4} )^{3} \)
29 \( ( 1 - T^{3} + T^{6} )^{2} \)
31 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
37 \( ( 1 - T^{2} + T^{4} )^{3} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( ( 1 - T^{2} + T^{4} )^{3} \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( ( 1 + T^{2} )^{6} \)
89 \( ( 1 - T^{2} + T^{4} )^{3} \)
97 \( ( 1 - T^{2} + T^{4} )^{3} \)
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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

−6.74531088779858849272441276528, −6.39017644833699576671827921051, −6.32815259831209767459582453820, −5.91968013811819827111234448556, −5.86743904560511903108317603095, −5.83182621001742307512254369100, −5.76807462483322463207632117535, −5.60834362079694936214853210543, −5.27055425394881661938174309464, −4.94945502179875678007590050744, −4.67214313732893305457178922367, −4.58668668089972911401777969333, −4.24342911037498609944567819001, −4.20291965281873156129532022129, −3.87848955710667483028000241274, −3.76385080488174815014788069940, −3.61410886568110032811303808108, −3.49792566224398728540772887025, −2.99982777685841159506363002746, −2.50906662764242310474340886110, −2.47973191849619736266050775398, −2.25416813820693029840901358088, −1.98267904604144642443036249991, −1.64105632702645315374959372320, −1.30914334637195636000935397625, 1.30914334637195636000935397625, 1.64105632702645315374959372320, 1.98267904604144642443036249991, 2.25416813820693029840901358088, 2.47973191849619736266050775398, 2.50906662764242310474340886110, 2.99982777685841159506363002746, 3.49792566224398728540772887025, 3.61410886568110032811303808108, 3.76385080488174815014788069940, 3.87848955710667483028000241274, 4.20291965281873156129532022129, 4.24342911037498609944567819001, 4.58668668089972911401777969333, 4.67214313732893305457178922367, 4.94945502179875678007590050744, 5.27055425394881661938174309464, 5.60834362079694936214853210543, 5.76807462483322463207632117535, 5.83182621001742307512254369100, 5.86743904560511903108317603095, 5.91968013811819827111234448556, 6.32815259831209767459582453820, 6.39017644833699576671827921051, 6.74531088779858849272441276528

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

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