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Properties

Label 12-2888e6-1.1-c0e6-0-4
Degree $12$
Conductor $5.802\times 10^{20}$
Sign $1$
Analytic cond. $8.96449$
Root an. cond. $1.20054$
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  − 3·3-s − 8-s + 3·9-s + 3·24-s + 27-s + 6·41-s − 3·49-s + 6·59-s − 3·67-s − 3·72-s − 3·73-s − 6·81-s − 3·97-s + 3·107-s − 18·123-s + 127-s + 131-s + 137-s + 139-s + 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 18·177-s + ⋯
L(s)  = 1  − 3·3-s − 8-s + 3·9-s + 3·24-s + 27-s + 6·41-s − 3·49-s + 6·59-s − 3·67-s − 3·72-s − 3·73-s − 6·81-s − 3·97-s + 3·107-s − 18·123-s + 127-s + 131-s + 137-s + 139-s + 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 18·177-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(8.96449\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 19^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3677931352\)
\(L(\frac12)\) \(\approx\) \(0.3677931352\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.72341003911381127160521761349, −4.67416843674059025545600575245, −4.63864885002543198334428166261, −4.25174743023063751662454771625, −4.21770995754293184603222313351, −4.21432614299198279305933164858, −4.01387328541947606907394790053, −3.66733435266661490581066128895, −3.64229522992081764219625383405, −3.56306633726354533904744297740, −3.10406246297355204226742237205, −3.00591448165143880711705477898, −2.80241878517462114835139555339, −2.74211724122067830910684810425, −2.73123077938146838251121187694, −2.67884977988730874826374412182, −2.22434237898600641808525103009, −1.94653383442201932072962079994, −1.83720065416355188228658964347, −1.55461189943639786045468567311, −1.55290188225860965339383496743, −0.905599266671765704664414227366, −0.849039973070957189240485347789, −0.65478739918531091125600927708, −0.47606307668568991245867624131, 0.47606307668568991245867624131, 0.65478739918531091125600927708, 0.849039973070957189240485347789, 0.905599266671765704664414227366, 1.55290188225860965339383496743, 1.55461189943639786045468567311, 1.83720065416355188228658964347, 1.94653383442201932072962079994, 2.22434237898600641808525103009, 2.67884977988730874826374412182, 2.73123077938146838251121187694, 2.74211724122067830910684810425, 2.80241878517462114835139555339, 3.00591448165143880711705477898, 3.10406246297355204226742237205, 3.56306633726354533904744297740, 3.64229522992081764219625383405, 3.66733435266661490581066128895, 4.01387328541947606907394790053, 4.21432614299198279305933164858, 4.21770995754293184603222313351, 4.25174743023063751662454771625, 4.63864885002543198334428166261, 4.67416843674059025545600575245, 4.72341003911381127160521761349

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

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