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Properties

Label 12-497e6-1.1-c0e6-0-0
Degree $12$
Conductor $1.507\times 10^{16}$
Sign $1$
Analytic cond. $0.000232852$
Root an. cond. $0.498031$
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  − 2·2-s + 4-s − 7-s − 9-s − 2·11-s + 2·14-s + 2·18-s + 4·22-s − 2·23-s + 6·25-s − 28-s + 5·29-s − 36-s − 2·37-s + 5·43-s − 2·44-s + 4·46-s − 12·50-s − 2·53-s − 10·58-s + 63-s − 2·67-s − 71-s + 4·74-s + 2·77-s − 2·79-s − 10·86-s + ⋯
L(s)  = 1  − 2·2-s + 4-s − 7-s − 9-s − 2·11-s + 2·14-s + 2·18-s + 4·22-s − 2·23-s + 6·25-s − 28-s + 5·29-s − 36-s − 2·37-s + 5·43-s − 2·44-s + 4·46-s − 12·50-s − 2·53-s − 10·58-s + 63-s − 2·67-s − 71-s + 4·74-s + 2·77-s − 2·79-s − 10·86-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 71^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 71^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{6} \cdot 71^{6}\)
Sign: $1$
Analytic conductor: \(0.000232852\)
Root analytic conductor: \(0.498031\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{6} \cdot 71^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06515555644\)
\(L(\frac12)\) \(\approx\) \(0.06515555644\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
71 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
good2 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
3 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
5 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
29 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
43 \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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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

−6.28350311682062262145843488422, −5.95689316050752453985245421238, −5.89134472070633406731108176350, −5.67938563192093564520277274094, −5.55831798875631848935135975998, −5.35757705615206723546833232675, −4.95739576403266130538869642437, −4.83807365386159229648055091965, −4.71309350509782776565210941726, −4.67225288530080301600201497337, −4.55334420634347759943014634198, −4.17698302013461289314241021632, −4.08766076703942062243499734005, −3.60290950677281451055025368869, −3.38541029463958621504199682875, −3.07861153561539616322100472050, −2.90543274074206239647325310913, −2.80635597606420516492216060749, −2.74319740834892197712399436690, −2.71367136005282411962052943522, −2.29330482226314713418828637837, −1.88866092612823442478892388967, −1.20692342196864802818012926572, −1.16029882214857233151430911039, −0.65160028494637968019211226843, 0.65160028494637968019211226843, 1.16029882214857233151430911039, 1.20692342196864802818012926572, 1.88866092612823442478892388967, 2.29330482226314713418828637837, 2.71367136005282411962052943522, 2.74319740834892197712399436690, 2.80635597606420516492216060749, 2.90543274074206239647325310913, 3.07861153561539616322100472050, 3.38541029463958621504199682875, 3.60290950677281451055025368869, 4.08766076703942062243499734005, 4.17698302013461289314241021632, 4.55334420634347759943014634198, 4.67225288530080301600201497337, 4.71309350509782776565210941726, 4.83807365386159229648055091965, 4.95739576403266130538869642437, 5.35757705615206723546833232675, 5.55831798875631848935135975998, 5.67938563192093564520277274094, 5.89134472070633406731108176350, 5.95689316050752453985245421238, 6.28350311682062262145843488422

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

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