Properties

Label 8-966e4-1.1-c1e4-0-2
Degree $8$
Conductor $870780120336$
Sign $1$
Analytic cond. $3540.11$
Root an. cond. $2.77732$
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  − 4·2-s + 10·4-s − 20·8-s − 2·9-s + 35·16-s + 8·18-s − 12·23-s − 20·25-s − 32·29-s − 56·32-s − 20·36-s + 48·46-s + 80·50-s + 128·58-s + 84·64-s − 32·71-s + 40·72-s + 3·81-s − 120·92-s − 200·100-s − 320·116-s + 16·121-s + 127-s − 120·128-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2.82·2-s + 5·4-s − 7.07·8-s − 2/3·9-s + 35/4·16-s + 1.88·18-s − 2.50·23-s − 4·25-s − 5.94·29-s − 9.89·32-s − 3.33·36-s + 7.07·46-s + 11.3·50-s + 16.8·58-s + 21/2·64-s − 3.79·71-s + 4.71·72-s + 1/3·81-s − 12.5·92-s − 20·100-s − 29.7·116-s + 1.45·121-s + 0.0887·127-s − 10.6·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(3540.11\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.08420031402\)
\(L(\frac12)\) \(\approx\) \(0.08420031402\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 + T )^{4} \)
3$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_2^2$ \( 1 + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 60 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 - p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 144 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 152 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−7.38330976588463643333262110134, −7.13967926797188617202445602979, −7.13395284651881474799843090172, −6.37834256984472070355628159559, −6.34474618527743319075033249532, −6.10484694106216092854733672954, −6.02180545433059201664409736172, −5.58563532061931022759432550801, −5.54338312246693281725493174865, −5.52351381928170091450649863454, −5.18118586072934107701340580734, −4.36699355013018486580527440526, −4.11536461193976544068174352103, −3.96019175591788892566974230340, −3.93240019432148899056232443315, −3.44616253786682771366573110760, −3.07534140117014699355938283479, −3.01871155424124785698321771280, −2.26299169259962734469761575922, −2.06103856419450853284310724767, −1.90231437888449660526363786799, −1.73228867107146031092768949758, −1.58173653741041645162213418788, −0.41855633493284544273849652999, −0.21838452420015866643731914774, 0.21838452420015866643731914774, 0.41855633493284544273849652999, 1.58173653741041645162213418788, 1.73228867107146031092768949758, 1.90231437888449660526363786799, 2.06103856419450853284310724767, 2.26299169259962734469761575922, 3.01871155424124785698321771280, 3.07534140117014699355938283479, 3.44616253786682771366573110760, 3.93240019432148899056232443315, 3.96019175591788892566974230340, 4.11536461193976544068174352103, 4.36699355013018486580527440526, 5.18118586072934107701340580734, 5.52351381928170091450649863454, 5.54338312246693281725493174865, 5.58563532061931022759432550801, 6.02180545433059201664409736172, 6.10484694106216092854733672954, 6.34474618527743319075033249532, 6.37834256984472070355628159559, 7.13395284651881474799843090172, 7.13967926797188617202445602979, 7.38330976588463643333262110134

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