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Properties

Label 8-440e4-1.1-c0e4-0-2
Degree $8$
Conductor $37480960000$
Sign $1$
Analytic cond. $0.00232508$
Root an. cond. $0.468602$
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  − 2-s − 5-s + 3·7-s − 9-s + 10-s − 11-s + 3·13-s − 3·14-s + 18-s − 2·19-s + 22-s − 2·23-s − 3·26-s + 32-s − 3·35-s − 2·37-s + 2·38-s + 3·41-s + 45-s + 2·46-s − 2·47-s + 6·49-s − 2·53-s + 55-s − 2·59-s − 3·63-s − 64-s + ⋯
L(s)  = 1  − 2-s − 5-s + 3·7-s − 9-s + 10-s − 11-s + 3·13-s − 3·14-s + 18-s − 2·19-s + 22-s − 2·23-s − 3·26-s + 32-s − 3·35-s − 2·37-s + 2·38-s + 3·41-s + 45-s + 2·46-s − 2·47-s + 6·49-s − 2·53-s + 55-s − 2·59-s − 3·63-s − 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(0.00232508\)
Root analytic conductor: \(0.468602\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 11^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2627252662\)
\(L(\frac12)\) \(\approx\) \(0.2627252662\)
\(L(1)\) 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)$
bad2$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good3$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
7$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
29$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
53$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−8.183983189476524650027166011426, −8.048381792993070636340859293257, −8.035039239007306342141434803689, −7.86416696857003306102926389911, −7.78146330002105350293180902704, −6.96644416191614110142747530473, −6.95696077081742688410481928523, −6.74761370673886706917019624374, −6.08895901824075764272385281885, −6.07227384501106205875941073052, −5.66630941715995649230358771886, −5.59965569299876557005614545241, −5.56492710492019753904178142926, −4.68677146207509353573017999424, −4.65731196269482711007823960314, −4.30406733288640467429795235764, −4.16650527135973297854551414472, −4.16355447724190280730310117905, −3.35739528430371123996751629381, −3.14725705661514812305844321937, −2.95940450687167707467735651702, −1.90926833830897970862286288612, −1.90361583526390890951689268277, −1.88027193216260198049203875648, −0.942359534289069936298232069661, 0.942359534289069936298232069661, 1.88027193216260198049203875648, 1.90361583526390890951689268277, 1.90926833830897970862286288612, 2.95940450687167707467735651702, 3.14725705661514812305844321937, 3.35739528430371123996751629381, 4.16355447724190280730310117905, 4.16650527135973297854551414472, 4.30406733288640467429795235764, 4.65731196269482711007823960314, 4.68677146207509353573017999424, 5.56492710492019753904178142926, 5.59965569299876557005614545241, 5.66630941715995649230358771886, 6.07227384501106205875941073052, 6.08895901824075764272385281885, 6.74761370673886706917019624374, 6.95696077081742688410481928523, 6.96644416191614110142747530473, 7.78146330002105350293180902704, 7.86416696857003306102926389911, 8.035039239007306342141434803689, 8.048381792993070636340859293257, 8.183983189476524650027166011426

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