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Properties

Label 12-740e6-1.1-c0e6-0-1
Degree $12$
Conductor $1.642\times 10^{17}$
Sign $1$
Analytic cond. $0.00253707$
Root an. cond. $0.607707$
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  + 3·5-s + 8-s − 3·17-s + 3·25-s + 3·40-s − 3·41-s + 6·61-s − 9·85-s − 6·89-s + 3·109-s − 3·121-s − 2·125-s + 127-s + 131-s − 3·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 3·5-s + 8-s − 3·17-s + 3·25-s + 3·40-s − 3·41-s + 6·61-s − 9·85-s − 6·89-s + 3·109-s − 3·121-s − 2·125-s + 127-s + 131-s − 3·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 5^{6} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(0.00253707\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 5^{6} \cdot 37^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.102150144\)
\(L(\frac12)\) \(\approx\) \(1.102150144\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
5 \( ( 1 - T + T^{2} )^{3} \)
37 \( 1 - T^{3} + T^{6} \)
good3 \( 1 - T^{6} + T^{12} \)
7 \( 1 - T^{6} + T^{12} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 + T^{3} + T^{6} )^{2} \)
17 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
19 \( 1 - T^{6} + T^{12} \)
23 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 + T^{2} )^{6} \)
41 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T )^{6}( 1 + T )^{6} \)
47 \( ( 1 - T^{2} + T^{4} )^{3} \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( 1 - T^{6} + T^{12} \)
61 \( ( 1 - T )^{6}( 1 - T^{3} + T^{6} ) \)
67 \( 1 - T^{6} + T^{12} \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
79 \( 1 - T^{6} + T^{12} \)
83 \( 1 - T^{6} + T^{12} \)
89 \( ( 1 + T )^{6}( 1 + T^{3} + T^{6} ) \)
97 \( ( 1 - T^{3} + T^{6} )^{2} \)
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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

−5.78184061748954251746705590636, −5.61485890846033672625889973578, −5.36526090952423265772870701422, −5.32093266356875504372342506700, −5.20597500045679051619337218062, −5.20355590886950247572890592384, −4.68165216082251144984896465477, −4.49579745965404289209315013255, −4.49121342644326724142916923218, −4.46734847800556296835178806351, −3.91270979372496683527280602980, −3.90796774350333351588931003695, −3.85688342002510037711994142165, −3.61558843365025395956700532658, −3.13415677324725465830577282192, −2.99085623785292930398020984764, −2.78615704285774934878430028107, −2.55686350423113919474846541430, −2.38362438932606538742168557452, −2.01970820442044330070834099099, −1.97997818290919168181820299939, −1.83843151239687506966261324352, −1.79135842120695127340907220926, −1.28087949568840184623890570712, −1.02071262683050848732097463924, 1.02071262683050848732097463924, 1.28087949568840184623890570712, 1.79135842120695127340907220926, 1.83843151239687506966261324352, 1.97997818290919168181820299939, 2.01970820442044330070834099099, 2.38362438932606538742168557452, 2.55686350423113919474846541430, 2.78615704285774934878430028107, 2.99085623785292930398020984764, 3.13415677324725465830577282192, 3.61558843365025395956700532658, 3.85688342002510037711994142165, 3.90796774350333351588931003695, 3.91270979372496683527280602980, 4.46734847800556296835178806351, 4.49121342644326724142916923218, 4.49579745965404289209315013255, 4.68165216082251144984896465477, 5.20355590886950247572890592384, 5.20597500045679051619337218062, 5.32093266356875504372342506700, 5.36526090952423265772870701422, 5.61485890846033672625889973578, 5.78184061748954251746705590636

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

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