Properties

Label 8-15e8-1.1-c4e4-0-2
Degree $8$
Conductor $2562890625$
Sign $1$
Analytic cond. $292622.$
Root an. cond. $4.82267$
Motivic weight $4$
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  − 504·11-s − 329·16-s − 2.60e3·31-s − 1.26e4·41-s − 2.31e4·61-s − 2.04e4·71-s + 2.28e4·101-s + 1.00e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 1.65e5·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 4.16·11-s − 1.28·16-s − 2.70·31-s − 7.49·41-s − 6.21·61-s − 4.05·71-s + 2.24·101-s + 6.84·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.34e−5·173-s + 5.35·176-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + 2.01e−5·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(292622.\)
Root analytic conductor: \(4.82267\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.002138185340\)
\(L(\frac12)\) \(\approx\) \(0.002138185340\)
\(L(3)\) 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 \( 1 \)
5 \( 1 \)
good2$C_2^3$ \( 1 + 329 T^{4} + p^{16} T^{8} \)
7$C_2^3$ \( 1 - 7111198 T^{4} + p^{16} T^{8} \)
11$C_2$ \( ( 1 + 126 T + p^{4} T^{2} )^{4} \)
13$C_2^3$ \( 1 - 1302619486 T^{4} + p^{16} T^{8} \)
17$C_2^3$ \( 1 - 133512382 T^{4} + p^{16} T^{8} \)
19$C_2^2$ \( ( 1 - 256798 T^{2} + p^{8} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 151272063362 T^{4} + p^{16} T^{8} \)
29$C_2^2$ \( ( 1 + 265054 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 650 T + p^{4} T^{2} )^{4} \)
37$C_2^3$ \( 1 - 1642456827358 T^{4} + p^{16} T^{8} \)
41$C_2$ \( ( 1 + 3150 T + p^{4} T^{2} )^{4} \)
43$C_2^3$ \( 1 - 22531122583102 T^{4} + p^{16} T^{8} \)
47$C_2^3$ \( 1 + 35229109920578 T^{4} + p^{16} T^{8} \)
53$C_2^3$ \( 1 - 90369826510078 T^{4} + p^{16} T^{8} \)
59$C_2^2$ \( ( 1 - 24195518 T^{2} + p^{8} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 5782 T + p^{4} T^{2} )^{4} \)
67$C_2^3$ \( 1 - 724831933536382 T^{4} + p^{16} T^{8} \)
71$C_2$ \( ( 1 + 72 p T + p^{4} T^{2} )^{4} \)
73$C_2^3$ \( 1 + 16791040992962 T^{4} + p^{16} T^{8} \)
79$C_2^2$ \( ( 1 - 71799262 T^{2} + p^{8} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 3708377573377154 T^{4} + p^{16} T^{8} \)
89$C_2^2$ \( ( 1 - 124356638 T^{2} + p^{8} T^{4} )^{2} \)
97$C_2^3$ \( 1 + 6777796734182978 T^{4} + p^{16} T^{8} \)
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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

−8.236939576822532525904015451546, −8.001413353054781651855505821154, −7.72704110018025414752065254378, −7.39857102521711232765998062462, −7.28492797867781763682568460608, −6.91733046046314980654817511112, −6.84928218200705188487274217078, −6.29021040478947080666852137481, −5.79481542997291438200274128365, −5.78604939165953174994424769418, −5.52535323271009920402198717624, −4.97371417671952871869925077896, −4.80430472328005551342448989201, −4.76267430746406656164794591676, −4.62527454204075695807900430819, −3.65975757167091870198037519205, −3.43650138662076897022217161516, −3.15689903233654401698360164108, −2.88363102684553699960662801936, −2.55861670069233308514166072699, −1.93782618587624118618432793180, −1.67677521468806568350362145596, −1.59282506370957048088407604154, −0.07617712722087053367727273925, −0.06271255710838321324271480802, 0.06271255710838321324271480802, 0.07617712722087053367727273925, 1.59282506370957048088407604154, 1.67677521468806568350362145596, 1.93782618587624118618432793180, 2.55861670069233308514166072699, 2.88363102684553699960662801936, 3.15689903233654401698360164108, 3.43650138662076897022217161516, 3.65975757167091870198037519205, 4.62527454204075695807900430819, 4.76267430746406656164794591676, 4.80430472328005551342448989201, 4.97371417671952871869925077896, 5.52535323271009920402198717624, 5.78604939165953174994424769418, 5.79481542997291438200274128365, 6.29021040478947080666852137481, 6.84928218200705188487274217078, 6.91733046046314980654817511112, 7.28492797867781763682568460608, 7.39857102521711232765998062462, 7.72704110018025414752065254378, 8.001413353054781651855505821154, 8.236939576822532525904015451546

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