Properties

Label 8-275e4-1.1-c1e4-0-3
Degree $8$
Conductor $5719140625$
Sign $1$
Analytic cond. $23.2508$
Root an. cond. $1.48185$
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  + 2·4-s − 4·9-s + 4·11-s + 3·16-s − 8·29-s − 8·36-s + 24·41-s + 8·44-s + 20·49-s + 16·59-s + 8·61-s + 12·64-s − 16·79-s − 6·81-s + 8·89-s − 16·99-s − 8·101-s + 8·109-s − 16·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + ⋯
L(s)  = 1  + 4-s − 4/3·9-s + 1.20·11-s + 3/4·16-s − 1.48·29-s − 4/3·36-s + 3.74·41-s + 1.20·44-s + 20/7·49-s + 2.08·59-s + 1.02·61-s + 3/2·64-s − 1.80·79-s − 2/3·81-s + 0.847·89-s − 1.60·99-s − 0.796·101-s + 0.766·109-s − 1.48·116-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(23.2508\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{275} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.407654659\)
\(L(\frac12)\) \(\approx\) \(2.407654659\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
11$C_1$ \( ( 1 - T )^{4} \)
good2$D_4\times C_2$ \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \)
7$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 4 T^{2} - 170 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 76 T^{2} + 2454 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 92 T^{2} + 6486 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 244 T^{2} + 25030 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 316 T^{2} + 43270 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \)
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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

−8.609024649444682574509224626087, −8.478838844715940628802910640840, −8.454979523558487280598901720878, −7.70147510800215435011818007093, −7.44566097885244933517184743280, −7.39265919229779502224473295362, −7.26990349252116302854572442786, −6.89395966359866780253505746893, −6.46973436054866681085470473148, −6.25088904946254021683436791533, −6.02259013673371578289011117629, −5.72171463244198737573255066967, −5.53736346430189683680820110309, −5.40206477674209174978974510786, −4.86351319736955812035021954396, −4.42235240349404129175085364339, −3.99854168546424015710266382145, −3.80125312101073682934274451959, −3.73815666774460920540588790787, −3.02196510664762819355575242138, −2.67464430784910526810956774364, −2.32687646222128897370739740071, −2.21394959974488844257857504394, −1.33719664321169824589643543215, −0.824723059246608476425168087687, 0.824723059246608476425168087687, 1.33719664321169824589643543215, 2.21394959974488844257857504394, 2.32687646222128897370739740071, 2.67464430784910526810956774364, 3.02196510664762819355575242138, 3.73815666774460920540588790787, 3.80125312101073682934274451959, 3.99854168546424015710266382145, 4.42235240349404129175085364339, 4.86351319736955812035021954396, 5.40206477674209174978974510786, 5.53736346430189683680820110309, 5.72171463244198737573255066967, 6.02259013673371578289011117629, 6.25088904946254021683436791533, 6.46973436054866681085470473148, 6.89395966359866780253505746893, 7.26990349252116302854572442786, 7.39265919229779502224473295362, 7.44566097885244933517184743280, 7.70147510800215435011818007093, 8.454979523558487280598901720878, 8.478838844715940628802910640840, 8.609024649444682574509224626087

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