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Properties

Label 12-58e12-1.1-c0e6-0-0
Degree $12$
Conductor $1.449\times 10^{21}$
Sign $1$
Analytic cond. $22.3912$
Root an. cond. $1.29570$
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-s − 2·5-s − 9-s + 2·10-s − 2·13-s − 2·17-s + 18-s + 25-s + 2·26-s + 2·34-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 50-s − 2·53-s − 2·61-s + 4·65-s − 2·73-s + 2·74-s + 2·82-s + 4·85-s − 2·89-s − 2·90-s − 2·97-s + 98-s − 2·101-s + ⋯
L(s)  = 1  − 2-s − 2·5-s − 9-s + 2·10-s − 2·13-s − 2·17-s + 18-s + 25-s + 2·26-s + 2·34-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 50-s − 2·53-s − 2·61-s + 4·65-s − 2·73-s + 2·74-s + 2·82-s + 4·85-s − 2·89-s − 2·90-s − 2·97-s + 98-s − 2·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(22.3912\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 29^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001025088628\)
\(L(\frac12)\) \(\approx\) \(0.001025088628\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
29 \( 1 \)
good3 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
41 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
59 \( ( 1 - T )^{6}( 1 + T )^{6} \)
61 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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

−4.60857922638759206007218371242, −4.40662176625515689195603263092, −4.40068654994892696309010876951, −4.25600065812498450404180649119, −4.17096993838606355326764815271, −3.92946201051589895232279755522, −3.80355958531023798282039366291, −3.46368601955526968210096214485, −3.46053538549831930717725138565, −3.45552595057050463494415096464, −3.17911458557812779090465778757, −2.97006395959048560845532016404, −2.88674984978711218295384021648, −2.68217850969339098091388606376, −2.56301078166903564633845327095, −2.41156502618669270674940815144, −2.31142678917888882130571521242, −1.86846279474763692224480976999, −1.67688352348328040367087351093, −1.61333100877314964364469782377, −1.59795866858031388158649661879, −1.29477823899248074335395728317, −0.791797593988843908331206252063, −0.26308499647106882150646856266, −0.03967609095332729444969283358, 0.03967609095332729444969283358, 0.26308499647106882150646856266, 0.791797593988843908331206252063, 1.29477823899248074335395728317, 1.59795866858031388158649661879, 1.61333100877314964364469782377, 1.67688352348328040367087351093, 1.86846279474763692224480976999, 2.31142678917888882130571521242, 2.41156502618669270674940815144, 2.56301078166903564633845327095, 2.68217850969339098091388606376, 2.88674984978711218295384021648, 2.97006395959048560845532016404, 3.17911458557812779090465778757, 3.45552595057050463494415096464, 3.46053538549831930717725138565, 3.46368601955526968210096214485, 3.80355958531023798282039366291, 3.92946201051589895232279755522, 4.17096993838606355326764815271, 4.25600065812498450404180649119, 4.40068654994892696309010876951, 4.40662176625515689195603263092, 4.60857922638759206007218371242

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

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