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Properties

Label 12-228e6-1.1-c0e6-0-0
Degree $12$
Conductor $1.405\times 10^{14}$
Sign $1$
Analytic cond. $2.17046\times 10^{-6}$
Root an. cond. $0.337323$
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  − 3·13-s − 3·19-s − 27-s − 3·43-s + 6·61-s − 3·67-s − 3·73-s + 6·79-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 3·13-s − 3·19-s − 27-s − 3·43-s + 6·61-s − 3·67-s − 3·73-s + 6·79-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(2.17046\times 10^{-6}\)
Root analytic conductor: \(0.337323\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1266274446\)
\(L(\frac12)\) \(\approx\) \(0.1266274446\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T^{3} + T^{6} \)
19 \( ( 1 + T + T^{2} )^{3} \)
good5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 + T^{3} + T^{6} )^{2} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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 \( ( 1 + T^{3} + T^{6} )^{2} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
89 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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

−7.00352180687488904356497950034, −6.73920555100069961042905188598, −6.65931453732311154453761455005, −6.37067722750190494941192947119, −6.28481136834883854916484166649, −6.22429988363595210209764897536, −6.11849197904606841331446093956, −5.39321199046958206380717898193, −5.28333195368743484907088279275, −5.27501758696737679722278801814, −5.06868025867360532327881455005, −4.95923144033880540540075629516, −4.78179838998601791622405907564, −4.34081692186315113372638408770, −4.02634525145321782775232691616, −3.98117740043583356945384581605, −3.97661258265047751708350429215, −3.46422594205337024592826052817, −3.30686353550851249774019302440, −2.76015719602775940537678612997, −2.55699441501552719531567795139, −2.33882128117379877062134696589, −2.27874071348170025039357565269, −1.82463447928634437616145705469, −1.50982751497776689218005851643, 1.50982751497776689218005851643, 1.82463447928634437616145705469, 2.27874071348170025039357565269, 2.33882128117379877062134696589, 2.55699441501552719531567795139, 2.76015719602775940537678612997, 3.30686353550851249774019302440, 3.46422594205337024592826052817, 3.97661258265047751708350429215, 3.98117740043583356945384581605, 4.02634525145321782775232691616, 4.34081692186315113372638408770, 4.78179838998601791622405907564, 4.95923144033880540540075629516, 5.06868025867360532327881455005, 5.27501758696737679722278801814, 5.28333195368743484907088279275, 5.39321199046958206380717898193, 6.11849197904606841331446093956, 6.22429988363595210209764897536, 6.28481136834883854916484166649, 6.37067722750190494941192947119, 6.65931453732311154453761455005, 6.73920555100069961042905188598, 7.00352180687488904356497950034

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

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