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Properties

Label 8-273e4-1.1-c0e4-0-2
Degree $8$
Conductor $5554571841$
Sign $1$
Analytic cond. $0.000344571$
Root an. cond. $0.369113$
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  + 9-s − 2·16-s − 2·19-s − 2·31-s + 2·37-s − 6·43-s + 49-s + 2·67-s + 2·73-s + 2·97-s − 2·103-s + 4·109-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 2·171-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 9-s − 2·16-s − 2·19-s − 2·31-s + 2·37-s − 6·43-s + 49-s + 2·67-s + 2·73-s + 2·97-s − 2·103-s + 4·109-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 2·171-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(0.000344571\)
Root analytic conductor: \(0.369113\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3217960122\)
\(L(\frac12)\) \(\approx\) \(0.3217960122\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2^2$ \( 1 - T^{2} + T^{4} \)
7$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2^2$ \( 1 - T^{2} + T^{4} \)
good2$C_2^2$ \( ( 1 + T^{4} )^{2} \)
5$C_2^3$ \( 1 - T^{4} + T^{8} \)
11$C_2^3$ \( 1 - T^{4} + T^{8} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{4} \)
29$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
31$C_2$$\times$$C_2^2$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
37$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
41$C_2^3$ \( 1 - T^{4} + T^{8} \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \)
47$C_2^3$ \( 1 - T^{4} + T^{8} \)
53$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
67$C_2$$\times$$C_2^2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_2^3$ \( 1 - T^{4} + T^{8} \)
73$C_2$$\times$$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T^{2} )^{2} \)
79$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.995853668757567558734547247849, −8.458275221751799179445986034992, −8.421069920951592564501845676259, −8.308660176718113188897170199922, −8.107211141571774721641999490708, −7.46453818625146142633300934425, −7.19999517213636921111326628875, −7.19970698996099435722616349042, −6.82387818687788767839684934779, −6.67667278842671113982667147721, −6.19970708999153305762628923764, −6.13904334681480548249363222274, −5.97423186250830402451939335800, −5.17358557078102877837631555532, −4.93757266341048901716305037813, −4.81506723547575448029043494977, −4.79212920581174562751973967591, −4.03834894780255131311491278359, −3.87042298282541794138872455744, −3.69437790266692302866185565657, −3.26023046269095444679245470634, −2.59829944923730075654488430825, −2.12255267909996772894638217098, −2.07231612362859248253041835300, −1.50214803518883393025586550229, 1.50214803518883393025586550229, 2.07231612362859248253041835300, 2.12255267909996772894638217098, 2.59829944923730075654488430825, 3.26023046269095444679245470634, 3.69437790266692302866185565657, 3.87042298282541794138872455744, 4.03834894780255131311491278359, 4.79212920581174562751973967591, 4.81506723547575448029043494977, 4.93757266341048901716305037813, 5.17358557078102877837631555532, 5.97423186250830402451939335800, 6.13904334681480548249363222274, 6.19970708999153305762628923764, 6.67667278842671113982667147721, 6.82387818687788767839684934779, 7.19970698996099435722616349042, 7.19999517213636921111326628875, 7.46453818625146142633300934425, 8.107211141571774721641999490708, 8.308660176718113188897170199922, 8.421069920951592564501845676259, 8.458275221751799179445986034992, 8.995853668757567558734547247849

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