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Properties

Label 12-464e6-1.1-c0e6-0-0
Degree $12$
Conductor $9.979\times 10^{15}$
Sign $1$
Analytic cond. $0.000154188$
Root an. cond. $0.481213$
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  − 2·5-s + 9-s + 2·13-s + 25-s + 29-s − 2·45-s − 49-s − 5·53-s − 4·65-s − 7·73-s + 7·97-s + 5·109-s + 2·117-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 2·5-s + 9-s + 2·13-s + 25-s + 29-s − 2·45-s − 49-s − 5·53-s − 4·65-s − 7·73-s + 7·97-s + 5·109-s + 2·117-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 29^{6}\)
Sign: $1$
Analytic conductor: \(0.000154188\)
Root analytic conductor: \(0.481213\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 29^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3051366503\)
\(L(\frac12)\) \(\approx\) \(0.3051366503\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
good3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
5 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
11 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
19 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
31 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
43 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
47 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
53 \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \)
59 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 + T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
79 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
97 \( ( 1 - T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \)
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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

−6.14712464571352391271807278710, −6.02316098712091952352754539002, −5.91468257270114324229862133909, −5.67003715766323875651853004634, −5.61697200128180013948029879703, −5.61099054889988574705750470866, −4.91292441925598379828458181599, −4.63819467551144122413712803692, −4.62393965288545909637296763408, −4.58163277355962828918673706021, −4.44521995937081907759283803717, −4.38441383400037705433552191162, −4.24163006774049858932726312714, −3.65717426460142515016259374230, −3.35918039897071382182492785726, −3.29925527209955661333690873722, −3.28542803971442198605744812048, −3.24643916904011129373248026730, −3.21961009298886406096180816203, −2.27972392454528652231305889292, −2.27398501249979046632991941344, −1.90189787865064361434623738074, −1.64978589096834160937635466117, −1.21568928905692828278696240149, −1.07360676217051346767138200650, 1.07360676217051346767138200650, 1.21568928905692828278696240149, 1.64978589096834160937635466117, 1.90189787865064361434623738074, 2.27398501249979046632991941344, 2.27972392454528652231305889292, 3.21961009298886406096180816203, 3.24643916904011129373248026730, 3.28542803971442198605744812048, 3.29925527209955661333690873722, 3.35918039897071382182492785726, 3.65717426460142515016259374230, 4.24163006774049858932726312714, 4.38441383400037705433552191162, 4.44521995937081907759283803717, 4.58163277355962828918673706021, 4.62393965288545909637296763408, 4.63819467551144122413712803692, 4.91292441925598379828458181599, 5.61099054889988574705750470866, 5.61697200128180013948029879703, 5.67003715766323875651853004634, 5.91468257270114324229862133909, 6.02316098712091952352754539002, 6.14712464571352391271807278710

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

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