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Properties

Label 12-1859e6-1.1-c0e6-0-0
Degree $12$
Conductor $4.127\times 10^{19}$
Sign $1$
Analytic cond. $0.637703$
Root an. cond. $0.963203$
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-s − 3·4-s − 2·5-s + 9-s − 3·11-s − 3·12-s − 2·15-s + 3·16-s + 6·20-s + 23-s + 25-s − 2·31-s − 3·33-s − 3·36-s + 37-s + 9·44-s − 2·45-s − 2·47-s + 3·48-s − 3·49-s − 2·53-s + 6·55-s + 59-s + 6·60-s + 2·64-s − 6·67-s + 69-s + ⋯
L(s)  = 1  + 3-s − 3·4-s − 2·5-s + 9-s − 3·11-s − 3·12-s − 2·15-s + 3·16-s + 6·20-s + 23-s + 25-s − 2·31-s − 3·33-s − 3·36-s + 37-s + 9·44-s − 2·45-s − 2·47-s + 3·48-s − 3·49-s − 2·53-s + 6·55-s + 59-s + 6·60-s + 2·64-s − 6·67-s + 69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(11^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(0.637703\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 11^{6} \cdot 13^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01208028446\)
\(L(\frac12)\) \(\approx\) \(0.01208028446\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( ( 1 + T + T^{2} )^{3} \)
13 \( 1 \)
good2 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
3 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
5 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
23 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
29 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
37 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
43 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
47 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
59 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( ( 1 + T + T^{2} )^{6} \)
71 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
97 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
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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

−4.83208272559391528737845568005, −4.75104560669187817785022607203, −4.73549139601209432308764670041, −4.55701244258842335996175365646, −4.53653097180458638557719166258, −4.29894685086623867264432826421, −4.20420971249250642612988501231, −3.98445062448233371355586929358, −3.81650923369002207303425411643, −3.70602010995952034742027978370, −3.46096774660225036332750638928, −3.37906710768151041114299336926, −3.18705813254827142067393018524, −3.12710546530344206179339086507, −2.94770158982838451215935443044, −2.63194283799297154669178767017, −2.62958429339965577284823109184, −2.24263851967859374971117561667, −2.05963625945230601434157637005, −1.82559529387583970240012192689, −1.70675833011910832240145033310, −1.29158706325489900435117947455, −1.14993001659317504856812892757, −0.45652607919955912686014237564, −0.099316330147147217617698363089, 0.099316330147147217617698363089, 0.45652607919955912686014237564, 1.14993001659317504856812892757, 1.29158706325489900435117947455, 1.70675833011910832240145033310, 1.82559529387583970240012192689, 2.05963625945230601434157637005, 2.24263851967859374971117561667, 2.62958429339965577284823109184, 2.63194283799297154669178767017, 2.94770158982838451215935443044, 3.12710546530344206179339086507, 3.18705813254827142067393018524, 3.37906710768151041114299336926, 3.46096774660225036332750638928, 3.70602010995952034742027978370, 3.81650923369002207303425411643, 3.98445062448233371355586929358, 4.20420971249250642612988501231, 4.29894685086623867264432826421, 4.53653097180458638557719166258, 4.55701244258842335996175365646, 4.73549139601209432308764670041, 4.75104560669187817785022607203, 4.83208272559391528737845568005

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

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