Properties

Label 8-75e4-1.1-c4e4-0-3
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $3612.62$
Root an. cond. $2.78437$
Motivic weight $4$
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  + 408·11-s + 112·16-s − 4.17e3·31-s − 9.55e3·41-s + 1.09e4·61-s + 48·71-s − 729·81-s − 6.43e3·101-s + 4.54e4·121-s + ⋯
L(s)  = 1  + 3.37·11-s + 7/16·16-s − 4.34·31-s − 5.68·41-s + 2.95·61-s + 0.00952·71-s − 1/9·81-s − 0.630·101-s + 3.10·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(3612.62\)
Root analytic conductor: \(2.78437\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :2, 2, 2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.180425618\)
\(L(\frac12)\) \(\approx\) \(2.180425618\)
\(L(3)\) 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$C_2^2$ \( 1 + p^{6} T^{4} \)
5 \( 1 \)
good2$C_2^3$ \( 1 - 7 p^{4} T^{4} + p^{16} T^{8} \)
7$C_2^3$ \( 1 + 3954623 T^{4} + p^{16} T^{8} \)
11$C_2$ \( ( 1 - 102 T + p^{4} T^{2} )^{4} \)
13$C_2^3$ \( 1 - 1572094417 T^{4} + p^{16} T^{8} \)
17$C_2^3$ \( 1 - 13352733982 T^{4} + p^{16} T^{8} \)
19$C_2^2$ \( ( 1 + 142583 T^{2} + p^{8} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 17358781538 T^{4} + p^{16} T^{8} \)
29$C_2^2$ \( ( 1 - 852062 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 1043 T + p^{4} T^{2} )^{4} \)
37$C_2^3$ \( 1 - 3181908370942 T^{4} + p^{16} T^{8} \)
41$C_2$ \( ( 1 + 2388 T + p^{4} T^{2} )^{4} \)
43$C_2^3$ \( 1 + 2822181031823 T^{4} + p^{16} T^{8} \)
47$C_2^3$ \( 1 + 43764717185378 T^{4} + p^{16} T^{8} \)
53$C_2^3$ \( 1 - 122506018477822 T^{4} + p^{16} T^{8} \)
59$C_2^2$ \( ( 1 + 16597378 T^{2} + p^{8} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 2747 T + p^{4} T^{2} )^{4} \)
67$C_2^3$ \( 1 + 810305584702943 T^{4} + p^{16} T^{8} \)
71$C_2$ \( ( 1 - 12 T + p^{4} T^{2} )^{4} \)
73$C_2^3$ \( 1 - 247548955923262 T^{4} + p^{16} T^{8} \)
79$C_2^2$ \( ( 1 - 68103262 T^{2} + p^{8} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 4110298709321182 T^{4} + p^{16} T^{8} \)
89$C_2^2$ \( ( 1 - 125048882 T^{2} + p^{8} T^{4} )^{2} \)
97$C_2^3$ \( 1 - 6632003107453297 T^{4} + p^{16} T^{8} \)
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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

−9.873164364680172936733386238256, −9.836223024412388190798622061827, −9.232518569481714286546477411536, −9.019060110714075976661783162339, −8.933472278387499847795324963920, −8.706237951570507395257856993539, −8.239336417125776553134055610529, −7.78204518234550211011710430038, −7.54562081389889541651956128626, −6.86806074261324162430656967451, −6.76183677074104487916642405836, −6.65378637431928569057235284111, −6.45732476294178879920773813695, −5.51049907582659605660170344799, −5.41683561694104961787424984369, −5.28804736010353819486233474748, −4.55289275747846174592043060461, −3.78133424504161081806569278175, −3.76023378050674464320544261999, −3.73655194654708845588071784170, −3.03511076454315712416204307425, −1.98278077696800617269811892616, −1.55360821702272650589334171784, −1.46581584867100129177805365849, −0.36264027725873022688477607192, 0.36264027725873022688477607192, 1.46581584867100129177805365849, 1.55360821702272650589334171784, 1.98278077696800617269811892616, 3.03511076454315712416204307425, 3.73655194654708845588071784170, 3.76023378050674464320544261999, 3.78133424504161081806569278175, 4.55289275747846174592043060461, 5.28804736010353819486233474748, 5.41683561694104961787424984369, 5.51049907582659605660170344799, 6.45732476294178879920773813695, 6.65378637431928569057235284111, 6.76183677074104487916642405836, 6.86806074261324162430656967451, 7.54562081389889541651956128626, 7.78204518234550211011710430038, 8.239336417125776553134055610529, 8.706237951570507395257856993539, 8.933472278387499847795324963920, 9.019060110714075976661783162339, 9.232518569481714286546477411536, 9.836223024412388190798622061827, 9.873164364680172936733386238256

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