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Properties

Label 12-152e6-1.1-c0e6-0-0
Degree $12$
Conductor $1.233\times 10^{13}$
Sign $1$
Analytic cond. $1.90547\times 10^{-7}$
Root an. cond. $0.275423$
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  − 3·3-s − 8-s + 3·9-s + 3·24-s + 27-s − 3·41-s − 3·49-s − 3·59-s − 3·67-s − 3·72-s + 6·73-s − 6·81-s − 3·97-s + 3·107-s + 9·123-s + 127-s + 131-s + 137-s + 139-s + 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 9·177-s + ⋯
L(s)  = 1  − 3·3-s − 8-s + 3·9-s + 3·24-s + 27-s − 3·41-s − 3·49-s − 3·59-s − 3·67-s − 3·72-s + 6·73-s − 6·81-s − 3·97-s + 3·107-s + 9·123-s + 127-s + 131-s + 137-s + 139-s + 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 9·177-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(1.90547\times 10^{-7}\)
Root analytic conductor: \(0.275423\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02859347458\)
\(L(\frac12)\) \(\approx\) \(0.02859347458\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−7.39758519646285917014352048049, −7.13243471349478150365941830909, −7.10497112277540841963935825572, −6.77135032788205920240945245072, −6.51769894007172285748150378118, −6.46149551819120856720807159243, −6.42903125424806765413335232275, −5.95939940282955441739391670155, −5.83413913608135674393486940162, −5.80589266530993807496984830934, −5.78546712841777911210077602306, −5.21030569718209570977292750569, −4.94318360401979582091211444118, −4.91519720797899269806536893658, −4.80514628386457119246701430492, −4.80112177596242387302279731740, −4.19419389257057914864481064603, −3.81922436752524545879087043351, −3.60482179663070603773393341866, −3.20332236583521029241306119119, −3.04413068870740298625400270153, −2.93868752547714674847099105795, −2.28876432896270193808624426026, −1.81909768811786175071966088324, −1.39037118027808155937599020260, 1.39037118027808155937599020260, 1.81909768811786175071966088324, 2.28876432896270193808624426026, 2.93868752547714674847099105795, 3.04413068870740298625400270153, 3.20332236583521029241306119119, 3.60482179663070603773393341866, 3.81922436752524545879087043351, 4.19419389257057914864481064603, 4.80112177596242387302279731740, 4.80514628386457119246701430492, 4.91519720797899269806536893658, 4.94318360401979582091211444118, 5.21030569718209570977292750569, 5.78546712841777911210077602306, 5.80589266530993807496984830934, 5.83413913608135674393486940162, 5.95939940282955441739391670155, 6.42903125424806765413335232275, 6.46149551819120856720807159243, 6.51769894007172285748150378118, 6.77135032788205920240945245072, 7.10497112277540841963935825572, 7.13243471349478150365941830909, 7.39758519646285917014352048049

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

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