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Properties

Label 8-595e4-1.1-c0e4-0-0
Degree $8$
Conductor $125333700625$
Sign $1$
Analytic cond. $0.00777492$
Root an. cond. $0.544925$
Motivic weight $0$
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 − 4·11-s + 10·16-s − 4·29-s + 16·44-s − 20·64-s − 4·71-s − 4·79-s − 2·81-s − 4·109-s + 16·116-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 40·176-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 4·4-s − 4·11-s + 10·16-s − 4·29-s + 16·44-s − 20·64-s − 4·71-s − 4·79-s − 2·81-s − 4·109-s + 16·116-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 40·176-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 7^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(0.00777492\)
Root analytic conductor: \(0.544925\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 7^{4} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006518754047\)
\(L(\frac12)\) \(\approx\) \(0.006518754047\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
good2$C_2$ \( ( 1 + T^{2} )^{4} \)
3$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
31$C_2^2$ \( ( 1 + T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_2^2$ \( ( 1 + T^{4} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−7.86380657804579272836053808470, −7.75394880620284699888510025152, −7.71161545284990536731656410722, −7.47254041939038334518056080669, −7.11184448652881892255316077113, −7.06785484517721604186883057229, −6.27575617279081981496722981098, −5.75333278769899776754496859040, −5.73042299282190757867389871636, −5.61586758707162161618097458945, −5.46153615906139493174586345169, −5.43520509553983783039974244091, −4.82827045801829154003294777363, −4.72743234054808983161556392147, −4.59855347284396071552998000400, −4.24434555103823740034276703966, −3.96391880066902145342679056734, −3.75157907312926211754391531390, −3.36830047096451656233174696508, −3.08343296203714461743507447948, −2.68968995654678575470923767835, −2.59895230610057306180734589747, −1.55508375793091427513291739097, −1.49699539496304710817003797098, −0.10143899253577293048531083236, 0.10143899253577293048531083236, 1.49699539496304710817003797098, 1.55508375793091427513291739097, 2.59895230610057306180734589747, 2.68968995654678575470923767835, 3.08343296203714461743507447948, 3.36830047096451656233174696508, 3.75157907312926211754391531390, 3.96391880066902145342679056734, 4.24434555103823740034276703966, 4.59855347284396071552998000400, 4.72743234054808983161556392147, 4.82827045801829154003294777363, 5.43520509553983783039974244091, 5.46153615906139493174586345169, 5.61586758707162161618097458945, 5.73042299282190757867389871636, 5.75333278769899776754496859040, 6.27575617279081981496722981098, 7.06785484517721604186883057229, 7.11184448652881892255316077113, 7.47254041939038334518056080669, 7.71161545284990536731656410722, 7.75394880620284699888510025152, 7.86380657804579272836053808470

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