Properties

Label 8-882e4-1.1-c2e4-0-18
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $333591.$
Root an. cond. $4.90232$
Motivic weight $2$
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  − 4·4-s + 12·16-s + 92·25-s + 256·37-s + 176·43-s − 32·64-s + 480·67-s + 368·79-s − 368·100-s + 280·109-s + 468·121-s + 127-s + 131-s + 137-s + 139-s − 1.02e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 352·169-s − 704·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 4-s + 3/4·16-s + 3.67·25-s + 6.91·37-s + 4.09·43-s − 1/2·64-s + 7.16·67-s + 4.65·79-s − 3.67·100-s + 2.56·109-s + 3.86·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 6.91·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.08·169-s − 4.09·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(333591.\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(8.602708304\)
\(L(\frac12)\) \(\approx\) \(8.602708304\)
\(L(2)\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 234 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 176 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 690 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 1050 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 432 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 770 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 64 T + p^{2} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 2962 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 44 T + p^{2} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 206 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 5280 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 3038 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 4704 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 120 T + p^{2} T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 10010 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 5040 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 92 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 1234 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 15442 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 18096 T^{2} + p^{4} T^{4} )^{2} \)
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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

−7.34551252542406505638621817932, −6.53270812195965430283299446912, −6.51765581789409209756598516428, −6.49135433342551387115967289084, −6.26656363743895845733921610655, −5.85892779683274785563746997203, −5.74478191134501118672409423052, −5.36074127679982741914115330568, −5.15494788372360179783538372542, −4.79013568905701027944009123222, −4.77635180840864639505962931359, −4.51373838564450187056943290538, −4.26405032177997103244180749411, −3.92333577198126412740137709391, −3.75923726436234364632360657169, −3.48434341800496029820808240195, −3.11662978885447902695817070140, −2.60140568911097692391326393597, −2.47627188026566385276826774656, −2.43483088063024155219764390066, −2.10069955843459331158687924849, −0.983934921236531911442892866607, −0.847246425636990681161218393380, −0.805067473172015488345116749782, −0.792999614294037286584586582458, 0.792999614294037286584586582458, 0.805067473172015488345116749782, 0.847246425636990681161218393380, 0.983934921236531911442892866607, 2.10069955843459331158687924849, 2.43483088063024155219764390066, 2.47627188026566385276826774656, 2.60140568911097692391326393597, 3.11662978885447902695817070140, 3.48434341800496029820808240195, 3.75923726436234364632360657169, 3.92333577198126412740137709391, 4.26405032177997103244180749411, 4.51373838564450187056943290538, 4.77635180840864639505962931359, 4.79013568905701027944009123222, 5.15494788372360179783538372542, 5.36074127679982741914115330568, 5.74478191134501118672409423052, 5.85892779683274785563746997203, 6.26656363743895845733921610655, 6.49135433342551387115967289084, 6.51765581789409209756598516428, 6.53270812195965430283299446912, 7.34551252542406505638621817932

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