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Properties

Label 12-38e12-1.1-c0e6-0-0
Degree $12$
Conductor $9.066\times 10^{18}$
Sign $1$
Analytic cond. $0.140070$
Root an. cond. $0.848910$
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  + 3·7-s + 3·11-s + 6·49-s + 9·77-s − 6·83-s + 6·121-s − 2·125-s + 127-s + 131-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 + 227-s + 229-s + ⋯
L(s)  = 1  + 3·7-s + 3·11-s + 6·49-s + 9·77-s − 6·83-s + 6·121-s − 2·125-s + 127-s + 131-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 + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(0.140070\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 19^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.450370757\)
\(L(\frac12)\) \(\approx\) \(2.450370757\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−5.16782434056809346188552814760, −4.87090081792470773872671750546, −4.79951486384512001158566651136, −4.74564587462473195367746254399, −4.65806610128235004011945769301, −4.48780028805021023472260622569, −4.44584563969055642202848499214, −4.08591757090576855183888262331, −3.84703604604651010125958572552, −3.80771342514797373758668450991, −3.75130509486854263393211824391, −3.63255962710652532149526077022, −3.50713896686204169279568213297, −2.99790497160659598509919664440, −2.90531903490272118231621679504, −2.63241949588713765969114059666, −2.43271337586194057182518111319, −2.33897840339074086965967395856, −2.22966226946056225556464826614, −1.71911695996803843455250430254, −1.56033483914506031062019341689, −1.46007341096216204260244962391, −1.34147835326360555266191305555, −1.18261387442914511522637451506, −0.903241290934177371385519360749, 0.903241290934177371385519360749, 1.18261387442914511522637451506, 1.34147835326360555266191305555, 1.46007341096216204260244962391, 1.56033483914506031062019341689, 1.71911695996803843455250430254, 2.22966226946056225556464826614, 2.33897840339074086965967395856, 2.43271337586194057182518111319, 2.63241949588713765969114059666, 2.90531903490272118231621679504, 2.99790497160659598509919664440, 3.50713896686204169279568213297, 3.63255962710652532149526077022, 3.75130509486854263393211824391, 3.80771342514797373758668450991, 3.84703604604651010125958572552, 4.08591757090576855183888262331, 4.44584563969055642202848499214, 4.48780028805021023472260622569, 4.65806610128235004011945769301, 4.74564587462473195367746254399, 4.79951486384512001158566651136, 4.87090081792470773872671750546, 5.16782434056809346188552814760

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

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