Properties

Label 12-1215e6-1.1-c0e6-0-0
Degree $12$
Conductor $3.217\times 10^{18}$
Sign $1$
Analytic cond. $0.0497050$
Root an. cond. $0.778693$
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  − 3·5-s − 2·8-s + 3·25-s + 6·40-s + 3·47-s − 3·49-s + 64-s − 6·107-s + 3·113-s − 3·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 3·5-s − 2·8-s + 3·25-s + 6·40-s + 3·47-s − 3·49-s + 64-s − 6·107-s + 3·113-s − 3·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{30} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(0.0497050\)
Root analytic conductor: \(0.778693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{30} \cdot 5^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05242363017\)
\(L(\frac12)\) \(\approx\) \(0.05242363017\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 + T + T^{2} )^{3} \)
good2 \( ( 1 + T^{3} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
17 \( ( 1 + T^{3} + T^{6} )^{2} \)
19 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 + T^{3} + T^{6} )^{2} \)
29 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
43 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
47 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
53 \( ( 1 + T^{3} + T^{6} )^{2} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
61 \( ( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
71 \( ( 1 - T )^{6}( 1 + T )^{6} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 + T^{3} + T^{6} )^{2} \)
83 \( ( 1 + T^{3} + T^{6} )^{2} \)
89 \( ( 1 - T )^{6}( 1 + T )^{6} \)
97 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
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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

−5.57619555562649377205914239020, −5.08042239849510953222938732151, −4.91064720073111717441306637375, −4.77327750538101906887754852075, −4.73987223099865087273200788547, −4.53201975391963347938290264483, −4.41897988616347564626738899807, −4.12743097976949670739868866370, −4.01722195090846001145097387853, −3.76125756602634345653048913374, −3.73769458630966904567558343286, −3.70257827248639002704683079580, −3.32463952586796080624503708210, −3.31461914421157489832965251043, −3.14026499578317593771942932448, −2.77737061251395926004580582686, −2.72102587627716572398435657778, −2.68788986385754697996449066133, −2.15744692020729398935865247413, −2.14647678931587313488208304827, −1.89688052582169433740470844022, −1.38852610148519858640062378496, −1.17277772312568379790859808718, −0.915388144085387702778319729450, −0.17649504841220544373247996542, 0.17649504841220544373247996542, 0.915388144085387702778319729450, 1.17277772312568379790859808718, 1.38852610148519858640062378496, 1.89688052582169433740470844022, 2.14647678931587313488208304827, 2.15744692020729398935865247413, 2.68788986385754697996449066133, 2.72102587627716572398435657778, 2.77737061251395926004580582686, 3.14026499578317593771942932448, 3.31461914421157489832965251043, 3.32463952586796080624503708210, 3.70257827248639002704683079580, 3.73769458630966904567558343286, 3.76125756602634345653048913374, 4.01722195090846001145097387853, 4.12743097976949670739868866370, 4.41897988616347564626738899807, 4.53201975391963347938290264483, 4.73987223099865087273200788547, 4.77327750538101906887754852075, 4.91064720073111717441306637375, 5.08042239849510953222938732151, 5.57619555562649377205914239020

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

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