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Properties

Label 12-3675e6-1.1-c0e6-0-0
Degree $12$
Conductor $2.463\times 10^{21}$
Sign $1$
Analytic cond. $38.0615$
Root an. cond. $1.35427$
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  + 3-s − 4-s + 7-s − 12-s + 2·13-s − 2·19-s + 21-s − 28-s − 2·31-s − 5·37-s + 2·39-s + 2·43-s − 2·52-s − 2·57-s + 5·61-s + 2·67-s + 2·73-s + 2·76-s − 2·79-s − 84-s + 2·91-s − 2·93-s + 2·97-s + 2·103-s − 2·109-s − 5·111-s − 121-s + ⋯
L(s)  = 1  + 3-s − 4-s + 7-s − 12-s + 2·13-s − 2·19-s + 21-s − 28-s − 2·31-s − 5·37-s + 2·39-s + 2·43-s − 2·52-s − 2·57-s + 5·61-s + 2·67-s + 2·73-s + 2·76-s − 2·79-s − 84-s + 2·91-s − 2·93-s + 2·97-s + 2·103-s − 2·109-s − 5·111-s − 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(38.0615\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8362200440\)
\(L(\frac12)\) \(\approx\) \(0.8362200440\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
5 \( 1 \)
7 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
37 \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
61 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
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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

−4.55734921397655318005754215147, −4.52531547473738031462206821622, −4.48590865400581654201887257630, −3.89811609693108531398010720851, −3.86476398045857856088981434983, −3.81744160506172788956330169686, −3.74293225803057957933522799558, −3.68252130772896741240799091767, −3.52127665029728753713689053197, −3.41801816061645757006689863784, −3.30625927153921541995748134035, −3.20733912244943114536192787841, −2.65620848058655192149989126326, −2.42542573759956641223643708949, −2.41484046480525956893133550414, −2.34550978354850565962655940450, −2.32565721685889527945515477445, −2.00107475435831608386243076115, −1.94428138124315090512446567237, −1.56277421951064506259952808043, −1.44255319651818887632486468222, −1.20584441630367466329160060852, −1.12482546376002106100738845214, −0.810365645500264228235861230368, −0.23758149772377306683202636400, 0.23758149772377306683202636400, 0.810365645500264228235861230368, 1.12482546376002106100738845214, 1.20584441630367466329160060852, 1.44255319651818887632486468222, 1.56277421951064506259952808043, 1.94428138124315090512446567237, 2.00107475435831608386243076115, 2.32565721685889527945515477445, 2.34550978354850565962655940450, 2.41484046480525956893133550414, 2.42542573759956641223643708949, 2.65620848058655192149989126326, 3.20733912244943114536192787841, 3.30625927153921541995748134035, 3.41801816061645757006689863784, 3.52127665029728753713689053197, 3.68252130772896741240799091767, 3.74293225803057957933522799558, 3.81744160506172788956330169686, 3.86476398045857856088981434983, 3.89811609693108531398010720851, 4.48590865400581654201887257630, 4.52531547473738031462206821622, 4.55734921397655318005754215147

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

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