Properties

Label 8-55e4-1.1-c5e4-0-0
Degree $8$
Conductor $9150625$
Sign $1$
Analytic cond. $6054.70$
Root an. cond. $2.97003$
Motivic weight $5$
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  − 62·3-s + 1.92e3·9-s − 2.04e3·16-s − 1.96e3·23-s + 3.00e3·25-s − 4.45e4·27-s − 2.53e3·37-s + 4.94e4·47-s + 1.26e5·48-s − 6.96e4·53-s − 1.45e5·67-s + 1.21e5·69-s − 2.65e5·71-s − 1.86e5·75-s + 1.02e6·81-s + 3.26e5·97-s + 3.60e5·103-s + 1.57e5·111-s − 3.84e5·113-s + 3.22e5·121-s + 127-s + 131-s + 137-s + 139-s − 3.06e6·141-s − 3.93e6·144-s + 149-s + ⋯
L(s)  = 1  − 3.97·3-s + 7.90·9-s − 2·16-s − 0.773·23-s + 0.960·25-s − 11.7·27-s − 0.304·37-s + 3.26·47-s + 7.95·48-s − 3.40·53-s − 3.96·67-s + 3.07·69-s − 6.24·71-s − 3.81·75-s + 17.2·81-s + 3.52·97-s + 3.34·103-s + 1.21·111-s − 2.83·113-s + 2·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 12.9·141-s − 15.8·144-s + 3.69e−6·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(9150625\)    =    \(5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(6054.70\)
Root analytic conductor: \(2.97003\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 9150625,\ (\ :5/2, 5/2, 5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1393197761\)
\(L(\frac12)\) \(\approx\) \(0.1393197761\)
\(L(\frac{7}{2})\) 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$C_2^2$ \( 1 - 3001 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
good2$C_2$ \( ( 1 - p^{3} T + p^{5} T^{2} )^{2}( 1 + p^{3} T + p^{5} T^{2} )^{2} \)
3$C_2$$\times$$C_2^2$ \( ( 1 + 31 T + p^{5} T^{2} )^{2}( 1 + 475 T^{2} + p^{10} T^{4} ) \)
7$C_2^2$ \( ( 1 + p^{10} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + p^{10} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + p^{10} T^{4} )^{2} \)
19$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + 981 T + p^{5} T^{2} )^{2}( 1 - 11910325 T^{2} + p^{10} T^{4} ) \)
29$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 3192323 T^{2} + p^{10} T^{4} )^{2} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + 1267 T + p^{5} T^{2} )^{2}( 1 - 137082625 T^{2} + p^{10} T^{4} ) \)
41$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + p^{10} T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 - 24708 T + p^{5} T^{2} )^{2}( 1 + 151795250 T^{2} + p^{10} T^{4} ) \)
53$C_2$$\times$$C_2^2$ \( ( 1 + 34806 T + p^{5} T^{2} )^{2}( 1 + 375066650 T^{2} + p^{10} T^{4} ) \)
59$C_2$ \( ( 1 - 24825 T + p^{5} T^{2} )^{2}( 1 + 24825 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + 72917 T + p^{5} T^{2} )^{2}( 1 + 2616638675 T^{2} + p^{10} T^{4} ) \)
71$C_2$ \( ( 1 + 66273 T + p^{5} T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + p^{10} T^{4} )^{2} \)
79$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + p^{10} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 2870912977 T^{2} + p^{10} T^{4} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 - 163183 T + p^{5} T^{2} )^{2}( 1 + 9454010975 T^{2} + p^{10} T^{4} ) \)
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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

−10.44809324729893950189435982432, −10.32718121200992618957336514724, −10.25951942920231825248700164260, −9.298510069665641843960276193407, −9.145166513384579021897552131720, −9.106849533811250561300834339886, −8.541194125216861870886284988977, −7.74586168109525604892579869102, −7.47958120468085375775936831234, −7.19935782064137758785701602019, −6.97122692260467919454905722936, −6.31851605203211662268866111963, −6.03104392316960182183820863167, −5.92659161993122784020127345261, −5.89011433969331129648823448050, −5.21036468498612630645065562287, −4.60442939531113757148053882581, −4.59785915703742243021769629018, −4.46844795272248172565536789293, −3.53801941144480896568440461992, −2.78832256560050963407393932376, −1.88084474258109118464182466984, −1.42418039716598719991066393790, −0.51353528424857197713019073309, −0.23637982855565413530491008312, 0.23637982855565413530491008312, 0.51353528424857197713019073309, 1.42418039716598719991066393790, 1.88084474258109118464182466984, 2.78832256560050963407393932376, 3.53801941144480896568440461992, 4.46844795272248172565536789293, 4.59785915703742243021769629018, 4.60442939531113757148053882581, 5.21036468498612630645065562287, 5.89011433969331129648823448050, 5.92659161993122784020127345261, 6.03104392316960182183820863167, 6.31851605203211662268866111963, 6.97122692260467919454905722936, 7.19935782064137758785701602019, 7.47958120468085375775936831234, 7.74586168109525604892579869102, 8.541194125216861870886284988977, 9.106849533811250561300834339886, 9.145166513384579021897552131720, 9.298510069665641843960276193407, 10.25951942920231825248700164260, 10.32718121200992618957336514724, 10.44809324729893950189435982432

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