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Properties

Label 12-740e6-1.1-c0e6-0-0
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  − 8-s + 3·17-s − 3·41-s + 6·61-s − 6·89-s + 3·109-s − 3·121-s − 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 + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 8-s + 3·17-s − 3·41-s + 6·61-s − 6·89-s + 3·109-s − 3·121-s − 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 + 197-s + 199-s + 211-s + 223-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

\(L(\frac{1}{2})\) \(\approx\) \(0.6104976081\)
\(L(\frac12)\) \(\approx\) \(0.6104976081\)
\(L(1)\) 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 \( 1 + T^{3} + T^{6} \)
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

−5.73539464792741575759889800065, −5.67359775931478171483249365750, −5.53264832554845244854185194814, −5.27256204977372017062735375083, −5.21911406394387334013014812484, −4.95954238513295204738181253626, −4.77924383911978708641608774191, −4.67507523136334128696339798195, −4.59390829803435235018623090709, −3.96213871767236746525148993834, −3.95025139399694602682106769758, −3.78756894178174871938546719001, −3.68095173507179966243868181147, −3.52127325421071566060800911900, −3.51632225039932458220400031095, −2.95687397918705146117051880917, −2.89145377097158957552292732831, −2.74607418507797769735726675413, −2.51318433613646876594283736873, −2.44247085328623403121680753065, −1.88751046423456277461827290529, −1.66870753981870348408845414343, −1.52559792997459738973635535965, −1.10631269484521126706326111314, −0.880419456150846754441715314779, 0.880419456150846754441715314779, 1.10631269484521126706326111314, 1.52559792997459738973635535965, 1.66870753981870348408845414343, 1.88751046423456277461827290529, 2.44247085328623403121680753065, 2.51318433613646876594283736873, 2.74607418507797769735726675413, 2.89145377097158957552292732831, 2.95687397918705146117051880917, 3.51632225039932458220400031095, 3.52127325421071566060800911900, 3.68095173507179966243868181147, 3.78756894178174871938546719001, 3.95025139399694602682106769758, 3.96213871767236746525148993834, 4.59390829803435235018623090709, 4.67507523136334128696339798195, 4.77924383911978708641608774191, 4.95954238513295204738181253626, 5.21911406394387334013014812484, 5.27256204977372017062735375083, 5.53264832554845244854185194814, 5.67359775931478171483249365750, 5.73539464792741575759889800065

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

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