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Properties

Label 12-891e6-1.1-c0e6-0-0
Degree $12$
Conductor $5.003\times 10^{17}$
Sign $1$
Analytic cond. $0.00773052$
Root an. cond. $0.666833$
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 + 3·25-s − 3·31-s + 3·47-s + 3·59-s − 64-s + 6·67-s − 3·89-s + 6·97-s − 3·103-s − 6·113-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 9·155-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 3·5-s + 3·25-s − 3·31-s + 3·47-s + 3·59-s − 64-s + 6·67-s − 3·89-s + 6·97-s − 3·103-s − 6·113-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 9·155-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{24} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(0.00773052\)
Root analytic conductor: \(0.666833\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{24} \cdot 11^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.388512593\)
\(L(\frac12)\) \(\approx\) \(1.388512593\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−5.52852026498495991259852775057, −5.46213675241123822009675448769, −5.42174303193735835372853489734, −5.13769385283179558923184805591, −5.06285736144392407177726421376, −5.05461895669355857688658682327, −4.96931224746665663039682908092, −4.21705594140991751703657195526, −4.08197706415534682286472529825, −4.05516295271595656206440941316, −3.98658661167708060874010629588, −3.92555916246389190828391454642, −3.70720460187797399164957601600, −3.21914569680158891187957791847, −3.17822935935729494166662329983, −3.00742638445069124948313363457, −2.47501260027218162498245284403, −2.40734038026332924882172110955, −2.34512936955244387043449387680, −2.18348318160314582980697635026, −2.04804590067180112763213581919, −1.83862428285239712791282220009, −1.32085345870502991971871489209, −1.19634339915372808604990390571, −1.05350538390653493851884988145, 1.05350538390653493851884988145, 1.19634339915372808604990390571, 1.32085345870502991971871489209, 1.83862428285239712791282220009, 2.04804590067180112763213581919, 2.18348318160314582980697635026, 2.34512936955244387043449387680, 2.40734038026332924882172110955, 2.47501260027218162498245284403, 3.00742638445069124948313363457, 3.17822935935729494166662329983, 3.21914569680158891187957791847, 3.70720460187797399164957601600, 3.92555916246389190828391454642, 3.98658661167708060874010629588, 4.05516295271595656206440941316, 4.08197706415534682286472529825, 4.21705594140991751703657195526, 4.96931224746665663039682908092, 5.05461895669355857688658682327, 5.06285736144392407177726421376, 5.13769385283179558923184805591, 5.42174303193735835372853489734, 5.46213675241123822009675448769, 5.52852026498495991259852775057

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

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