Properties

Label 8-210e4-1.1-c3e4-0-1
Degree $8$
Conductor $1944810000$
Sign $1$
Analytic cond. $23569.0$
Root an. cond. $3.52000$
Motivic weight $3$
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  − 8·4-s − 4·5-s − 18·9-s + 24·11-s + 48·16-s + 8·19-s + 32·20-s + 146·25-s − 40·29-s − 584·31-s + 144·36-s − 264·41-s − 192·44-s + 72·45-s − 98·49-s − 96·55-s − 448·59-s − 24·61-s − 256·64-s − 312·71-s − 64·76-s − 752·79-s − 192·80-s + 243·81-s − 1.09e3·89-s − 32·95-s − 432·99-s + ⋯
L(s)  = 1  − 4-s − 0.357·5-s − 2/3·9-s + 0.657·11-s + 3/4·16-s + 0.0965·19-s + 0.357·20-s + 1.16·25-s − 0.256·29-s − 3.38·31-s + 2/3·36-s − 1.00·41-s − 0.657·44-s + 0.238·45-s − 2/7·49-s − 0.235·55-s − 0.988·59-s − 0.0503·61-s − 1/2·64-s − 0.521·71-s − 0.0965·76-s − 1.07·79-s − 0.268·80-s + 1/3·81-s − 1.30·89-s − 0.0345·95-s − 0.438·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(23569.0\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.07769183680\)
\(L(\frac12)\) \(\approx\) \(0.07769183680\)
\(L(\frac{5}{2})\) 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)$
bad2$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
3$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
5$C_2^2$ \( 1 + 4 T - 26 p T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
7$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
good11$D_{4}$ \( ( 1 - 12 T + 2314 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 716 T^{2} - 5578218 T^{4} - 716 p^{6} T^{6} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 - 9540 T^{2} + 48909638 T^{4} - 9540 p^{6} T^{6} + p^{12} T^{8} \)
19$D_{4}$ \( ( 1 - 4 T + 13626 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 14172 T^{2} + 279829670 T^{4} - 14172 p^{6} T^{6} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 20 T + 48782 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 292 T + 80514 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 122420 T^{2} + 7280062518 T^{4} - 122420 p^{6} T^{6} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 + 132 T + 136054 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 82316 T^{2} + 14216284662 T^{4} - 82316 p^{6} T^{6} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 - 198492 T^{2} + 22457318918 T^{4} - 198492 p^{6} T^{6} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 110828 T^{2} + 21717634998 T^{4} - 110828 p^{6} T^{6} + p^{12} T^{8} \)
59$D_{4}$ \( ( 1 + 224 T + 30086 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 12 T + 292622 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 487820 T^{2} + 116820910038 T^{4} - 487820 p^{6} T^{6} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 + 156 T + 491410 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 1463324 T^{2} + 835923760038 T^{4} - 1463324 p^{6} T^{6} + p^{12} T^{8} \)
79$D_{4}$ \( ( 1 + 376 T + 1015278 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 125228 T^{2} + 394047302358 T^{4} - 125228 p^{6} T^{6} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 + 548 T + 237398 T^{2} + 548 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 2031292 T^{2} + 2294372967174 T^{4} - 2031292 p^{6} T^{6} + p^{12} T^{8} \)
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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

−8.640292286518087446431438172109, −8.584882977934690859713367202028, −8.096823550570734575065821451752, −7.66775032373958656470285886458, −7.50987924174870906328609297680, −7.33105876885253735161966102512, −6.99844601033642910552607900615, −6.66667252634248949789953071324, −6.36452384848430569024430398647, −5.98592879194487554795898484134, −5.78281987564497763226458096840, −5.27295449425948314989095103606, −5.26276454794895947959319157823, −4.98250944213868018471642099689, −4.34746701194927140644055397673, −4.30879966739248561189560399959, −3.87389408965135190523375148902, −3.48644613800062838769438118577, −3.24716889690416571304161461678, −2.95676985804368448748491716539, −2.33240538198941657152457753491, −1.71959400382534641412037710552, −1.49091904509365853135453919702, −0.78274227703291363125779283803, −0.06981912432493711037619914447, 0.06981912432493711037619914447, 0.78274227703291363125779283803, 1.49091904509365853135453919702, 1.71959400382534641412037710552, 2.33240538198941657152457753491, 2.95676985804368448748491716539, 3.24716889690416571304161461678, 3.48644613800062838769438118577, 3.87389408965135190523375148902, 4.30879966739248561189560399959, 4.34746701194927140644055397673, 4.98250944213868018471642099689, 5.26276454794895947959319157823, 5.27295449425948314989095103606, 5.78281987564497763226458096840, 5.98592879194487554795898484134, 6.36452384848430569024430398647, 6.66667252634248949789953071324, 6.99844601033642910552607900615, 7.33105876885253735161966102512, 7.50987924174870906328609297680, 7.66775032373958656470285886458, 8.096823550570734575065821451752, 8.584882977934690859713367202028, 8.640292286518087446431438172109

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