Properties

Label 8-63e4-1.1-c1e4-0-0
Degree $8$
Conductor $15752961$
Sign $1$
Analytic cond. $0.0640428$
Root an. cond. $0.709265$
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 − 2·7-s + 4·16-s + 6·19-s + 4·25-s + 4·28-s + 6·31-s + 2·37-s − 4·43-s − 11·49-s − 12·61-s − 16·64-s − 22·67-s + 6·73-s − 12·76-s − 10·79-s − 8·100-s − 30·103-s + 2·109-s − 8·112-s − 20·121-s − 12·124-s + 127-s + 131-s − 12·133-s + 137-s + 139-s + ⋯
L(s)  = 1  − 4-s − 0.755·7-s + 16-s + 1.37·19-s + 4/5·25-s + 0.755·28-s + 1.07·31-s + 0.328·37-s − 0.609·43-s − 1.57·49-s − 1.53·61-s − 2·64-s − 2.68·67-s + 0.702·73-s − 1.37·76-s − 1.12·79-s − 4/5·100-s − 2.95·103-s + 0.191·109-s − 0.755·112-s − 1.81·121-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s − 1.04·133-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(15752961\)    =    \(3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.0640428\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{63} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 15752961,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4929819705\)
\(L(\frac12)\) \(\approx\) \(0.4929819705\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
good2$C_2$$\times$$C_2^2$ \( ( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} ) \)
5$C_2^3$ \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^3$ \( 1 + 20 T^{2} + 279 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^3$ \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
47$C_2^3$ \( 1 + 56 T^{2} + 927 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 98 T^{2} + 6795 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 94 T^{2} + 5355 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 6 T + 73 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
71$C_2^2$ \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
79$C_2^2$ \( ( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 154 T^{2} + 15795 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
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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

−11.10174018932143255756913055133, −10.70163208774571586404402313269, −10.65669465251123000905145826379, −10.22998312758584522794044048633, −9.930467654331260798011592151289, −9.480438361217717416520547736614, −9.411742933182240567692043448061, −9.195924563868058470989953106562, −8.823945871600427473843927325438, −8.164254192915018366618864657587, −8.088397365210213726868367655980, −7.988805816840212018400917385632, −7.23917560581803928233065140131, −6.97438849727951962623516571759, −6.76390461462736867402788506386, −6.11027014199763665193292161959, −5.72977333252804673059989193190, −5.68353393675272583983688301244, −4.86023199039674832031648035943, −4.62308331320787136533189059178, −4.37645112329284509211933291668, −3.56071807803503707916438360535, −3.05571607448660817101897693644, −2.95784791817886436634704789714, −1.47602106964119664343560575942, 1.47602106964119664343560575942, 2.95784791817886436634704789714, 3.05571607448660817101897693644, 3.56071807803503707916438360535, 4.37645112329284509211933291668, 4.62308331320787136533189059178, 4.86023199039674832031648035943, 5.68353393675272583983688301244, 5.72977333252804673059989193190, 6.11027014199763665193292161959, 6.76390461462736867402788506386, 6.97438849727951962623516571759, 7.23917560581803928233065140131, 7.988805816840212018400917385632, 8.088397365210213726868367655980, 8.164254192915018366618864657587, 8.823945871600427473843927325438, 9.195924563868058470989953106562, 9.411742933182240567692043448061, 9.480438361217717416520547736614, 9.930467654331260798011592151289, 10.22998312758584522794044048633, 10.65669465251123000905145826379, 10.70163208774571586404402313269, 11.10174018932143255756913055133

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