Properties

Label 8-975e4-1.1-c1e4-0-37
Degree $8$
Conductor $903687890625$
Sign $1$
Analytic cond. $3673.89$
Root an. cond. $2.79023$
Motivic weight $1$
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  + 2-s + 2·3-s + 2·6-s − 3·7-s − 5·8-s + 9-s + 4·11-s − 2·13-s − 3·14-s − 3·16-s + 17-s + 18-s − 6·19-s − 6·21-s + 4·22-s + 4·23-s − 10·24-s − 2·26-s − 2·27-s − 29-s + 2·31-s + 8·33-s + 34-s + 11·37-s − 6·38-s − 4·39-s − 41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 0.816·6-s − 1.13·7-s − 1.76·8-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.801·14-s − 3/4·16-s + 0.242·17-s + 0.235·18-s − 1.37·19-s − 1.30·21-s + 0.852·22-s + 0.834·23-s − 2.04·24-s − 0.392·26-s − 0.384·27-s − 0.185·29-s + 0.359·31-s + 1.39·33-s + 0.171·34-s + 1.80·37-s − 0.973·38-s − 0.640·39-s − 0.156·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3673.89\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.379771428\)
\(L(\frac12)\) \(\approx\) \(4.379771428\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
5 \( 1 \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 - T + T^{2} + p^{2} T^{3} - 3 p T^{4} + p^{3} T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 + 3 T - 3 T^{2} - 6 T^{3} + 32 T^{4} - 6 p T^{5} - 3 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - T - 29 T^{2} + 4 T^{3} + 594 T^{4} + 4 p T^{5} - 29 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 6 T + 6 T^{2} - 48 T^{3} - 145 T^{4} - 48 p T^{5} + 6 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 + T - 19 T^{2} - 38 T^{3} - 470 T^{4} - 38 p T^{5} - 19 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 11 T + 21 T^{2} - 286 T^{3} + 4154 T^{4} - 286 p T^{5} + 21 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 + T - 77 T^{2} - 4 T^{3} + 4362 T^{4} - 4 p T^{5} - 77 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 5 T - 63 T^{2} - 10 T^{3} + 4820 T^{4} - 10 p T^{5} - 63 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 11 T + 98 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 14 T + 46 T^{2} - 448 T^{3} + 7455 T^{4} - 448 p T^{5} + 46 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 16 T + 87 T^{2} + 752 T^{3} + 9224 T^{4} + 752 p T^{5} + 87 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 5 T - 111 T^{2} - 10 T^{3} + 12332 T^{4} - 10 p T^{5} - 111 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 14 T + 125 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 12 T + 165 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 15 T + 210 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 18 T + 82 T^{2} + 1152 T^{3} + 21807 T^{4} + 1152 p T^{5} + 82 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 13 T - 63 T^{2} + 494 T^{3} + 28022 T^{4} + 494 p T^{5} - 63 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.06739785474640266003685719145, −6.70783643648768649749139802353, −6.67212502271757291965837655012, −6.66957099178080529037733587259, −6.13544078829004955460232846590, −5.95013017843407542749611852900, −5.92715030788677605878989165740, −5.82538229698854230968467569620, −5.39213364821839529503591045620, −4.93296840062150340868412944574, −4.76313061121857128271091268688, −4.55412106691383929056307222794, −4.44311418178179329671002752566, −3.84808794721168710102820491845, −3.72383653396687762720485066821, −3.53603348775554680094542264816, −3.38277832236644816267337562490, −3.07267618811061767736544025787, −2.77334449714427897179832598158, −2.44628411343413607083807512047, −2.42124580708980020038662580469, −1.86473642305329737576385872953, −1.52368919756866851052166615222, −0.72059562148262381968354695796, −0.52148446564148092987501816469, 0.52148446564148092987501816469, 0.72059562148262381968354695796, 1.52368919756866851052166615222, 1.86473642305329737576385872953, 2.42124580708980020038662580469, 2.44628411343413607083807512047, 2.77334449714427897179832598158, 3.07267618811061767736544025787, 3.38277832236644816267337562490, 3.53603348775554680094542264816, 3.72383653396687762720485066821, 3.84808794721168710102820491845, 4.44311418178179329671002752566, 4.55412106691383929056307222794, 4.76313061121857128271091268688, 4.93296840062150340868412944574, 5.39213364821839529503591045620, 5.82538229698854230968467569620, 5.92715030788677605878989165740, 5.95013017843407542749611852900, 6.13544078829004955460232846590, 6.66957099178080529037733587259, 6.67212502271757291965837655012, 6.70783643648768649749139802353, 7.06739785474640266003685719145

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