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Properties

Label 12-344e6-1.1-c0e6-0-0
Degree $12$
Conductor $1.657\times 10^{15}$
Sign $1$
Analytic cond. $2.56031\times 10^{-5}$
Root an. cond. $0.414340$
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  − 2-s − 2·3-s + 2·6-s + 9-s − 2·11-s − 2·17-s − 18-s + 5·19-s + 2·22-s − 25-s + 4·33-s + 2·34-s − 5·38-s − 2·41-s − 43-s + 6·49-s + 50-s + 4·51-s − 10·57-s − 2·59-s − 4·66-s − 2·67-s − 2·73-s + 2·75-s + 2·82-s + 5·83-s + 86-s + ⋯
L(s)  = 1  − 2-s − 2·3-s + 2·6-s + 9-s − 2·11-s − 2·17-s − 18-s + 5·19-s + 2·22-s − 25-s + 4·33-s + 2·34-s − 5·38-s − 2·41-s − 43-s + 6·49-s + 50-s + 4·51-s − 10·57-s − 2·59-s − 4·66-s − 2·67-s − 2·73-s + 2·75-s + 2·82-s + 5·83-s + 86-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 43^{6}\)
Sign: $1$
Analytic conductor: \(2.56031\times 10^{-5}\)
Root analytic conductor: \(0.414340\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 43^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04432770572\)
\(L(\frac12)\) \(\approx\) \(0.04432770572\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
43 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
good3 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
7 \( ( 1 - T )^{6}( 1 + T )^{6} \)
11 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
19 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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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

−6.53125043285117827626466342279, −6.20001174570027212857977307040, −6.17822838415213605813131760136, −5.96178204035609687259641668065, −5.76798947606259489237411905089, −5.50650860779643453883056907470, −5.44875879939565007542378452577, −5.20798658969991922508014256657, −5.15448168191270438007927587333, −5.09244492943143590320286297250, −5.08123150497594287591359252110, −4.38264456019136927109038493175, −4.37661284175762954716656025174, −4.13378060126009624983329967457, −3.82713356824279677566174915286, −3.75231259594001000710590459664, −3.18072060962583270581830501526, −3.14140187504440430832521039740, −2.82949258839536539789104894818, −2.74102117807717627115368832369, −2.46539234798720148746249944003, −2.03497982295242738357036736634, −1.57917943088473393774603272829, −1.37756860400504313567997121253, −0.65104441533925965877777233190, 0.65104441533925965877777233190, 1.37756860400504313567997121253, 1.57917943088473393774603272829, 2.03497982295242738357036736634, 2.46539234798720148746249944003, 2.74102117807717627115368832369, 2.82949258839536539789104894818, 3.14140187504440430832521039740, 3.18072060962583270581830501526, 3.75231259594001000710590459664, 3.82713356824279677566174915286, 4.13378060126009624983329967457, 4.37661284175762954716656025174, 4.38264456019136927109038493175, 5.08123150497594287591359252110, 5.09244492943143590320286297250, 5.15448168191270438007927587333, 5.20798658969991922508014256657, 5.44875879939565007542378452577, 5.50650860779643453883056907470, 5.76798947606259489237411905089, 5.96178204035609687259641668065, 6.17822838415213605813131760136, 6.20001174570027212857977307040, 6.53125043285117827626466342279

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

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