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Properties

Label 12-21e12-1.1-c0e6-0-0
Degree $12$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $0.000113651$
Root an. cond. $0.469135$
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  + 4-s + 7-s − 25-s + 28-s − 5·37-s + 2·43-s − 7·61-s − 2·67-s − 2·79-s − 100-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 5·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 2·172-s + 173-s − 175-s + 179-s + ⋯
L(s)  = 1  + 4-s + 7-s − 25-s + 28-s − 5·37-s + 2·43-s − 7·61-s − 2·67-s − 2·79-s − 100-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 5·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 2·172-s + 173-s − 175-s + 179-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(0.000113651\)
Root analytic conductor: \(0.469135\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4761845287\)
\(L(\frac12)\) \(\approx\) \(0.4761845287\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
good2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
11 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
23 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
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 \( ( 1 + T )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} ) \)
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 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
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 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
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

−6.34730658325799405366575058053, −6.00368311096996551872152539710, −5.78104499938859262269552110800, −5.68000845334625560287929794526, −5.67292758131999990757562775390, −5.55360292523801343958950518386, −5.40500684954348814411753277659, −4.86668972540395718839892426619, −4.74069340594935601604084111127, −4.59019364907655445378541667039, −4.41969686715835042120173225413, −4.38668676347534611002566275752, −4.29872827720491438378431383270, −3.86018323134596211897653580691, −3.39025340258085313615061057248, −3.29510686126521703266866764348, −3.21070083437703102155310474852, −3.12611316205290176224731723765, −2.93796922257296819669021008800, −2.37718199879360827484788402084, −2.01762822689431474824084583115, −1.93272760017737609322555567365, −1.91746541672123797073516136363, −1.41866505169058123650939135130, −1.32729449626805128813890952013, 1.32729449626805128813890952013, 1.41866505169058123650939135130, 1.91746541672123797073516136363, 1.93272760017737609322555567365, 2.01762822689431474824084583115, 2.37718199879360827484788402084, 2.93796922257296819669021008800, 3.12611316205290176224731723765, 3.21070083437703102155310474852, 3.29510686126521703266866764348, 3.39025340258085313615061057248, 3.86018323134596211897653580691, 4.29872827720491438378431383270, 4.38668676347534611002566275752, 4.41969686715835042120173225413, 4.59019364907655445378541667039, 4.74069340594935601604084111127, 4.86668972540395718839892426619, 5.40500684954348814411753277659, 5.55360292523801343958950518386, 5.67292758131999990757562775390, 5.68000845334625560287929794526, 5.78104499938859262269552110800, 6.00368311096996551872152539710, 6.34730658325799405366575058053

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

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