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Properties

Label 12-399e6-1.1-c0e6-0-0
Degree $12$
Conductor $4.035\times 10^{15}$
Sign $1$
Analytic cond. $6.23419\times 10^{-5}$
Root an. cond. $0.446236$
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  − 3·13-s − 6·19-s + 27-s + 3·43-s + 3·61-s + 64-s − 3·67-s − 3·73-s + 3·79-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 3·13-s − 6·19-s + 27-s + 3·43-s + 3·61-s + 64-s − 3·67-s − 3·73-s + 3·79-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 7^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(6.23419\times 10^{-5}\)
Root analytic conductor: \(0.446236\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 7^{6} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2112081667\)
\(L(\frac12)\) \(\approx\) \(0.2112081667\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{3} + T^{6} \)
7 \( 1 + T^{3} + T^{6} \)
19 \( ( 1 + T )^{6} \)
good2 \( 1 - T^{6} + T^{12} \)
5 \( 1 - T^{6} + T^{12} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
17 \( 1 - T^{6} + T^{12} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( 1 - T^{6} + T^{12} \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
47 \( 1 - T^{6} + T^{12} \)
53 \( 1 - T^{6} + T^{12} \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 + T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
71 \( 1 - T^{6} + T^{12} \)
73 \( ( 1 + T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
79 \( ( 1 - T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{2} + T^{4} )^{3} \)
89 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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

−6.40642193289514643874027728686, −6.27230168895615760336101874052, −5.83839901651848611212724373460, −5.77914720872417108810956570008, −5.75863806697895405150878234097, −5.62912528322123446992134405007, −5.47743462409364528420023975247, −4.76908965403256997516232768925, −4.73715340245241738788994312315, −4.72555267104690265032080775721, −4.52557202555545190941733793829, −4.44589328010384454097979317401, −4.34528017148866353767443722678, −4.05368792781209756199380944555, −3.64424954936578271707490421861, −3.63853081672493994102938746524, −3.38567146211223645471376010990, −2.73316157442354504973908410888, −2.68684156072379915195060162685, −2.59864294242261779663412476605, −2.29563691214071385358234499522, −2.18763305166749446415079812959, −1.84901332212710527703579861260, −1.73637832239036442448336404241, −0.78169091650337380349031349174, 0.78169091650337380349031349174, 1.73637832239036442448336404241, 1.84901332212710527703579861260, 2.18763305166749446415079812959, 2.29563691214071385358234499522, 2.59864294242261779663412476605, 2.68684156072379915195060162685, 2.73316157442354504973908410888, 3.38567146211223645471376010990, 3.63853081672493994102938746524, 3.64424954936578271707490421861, 4.05368792781209756199380944555, 4.34528017148866353767443722678, 4.44589328010384454097979317401, 4.52557202555545190941733793829, 4.72555267104690265032080775721, 4.73715340245241738788994312315, 4.76908965403256997516232768925, 5.47743462409364528420023975247, 5.62912528322123446992134405007, 5.75863806697895405150878234097, 5.77914720872417108810956570008, 5.83839901651848611212724373460, 6.27230168895615760336101874052, 6.40642193289514643874027728686

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

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