Properties

Label 8-75e4-1.1-c2e4-0-0
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $17.4415$
Root an. cond. $1.42954$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 72·11-s + 7·16-s + 88·31-s − 72·41-s + 8·61-s + 288·71-s − 9·81-s − 432·101-s + 2.75e3·121-s + ⋯
L(s)  = 1  − 6.54·11-s + 7/16·16-s + 2.83·31-s − 1.75·41-s + 8/61·61-s + 4.05·71-s − 1/9·81-s − 4.27·101-s + 22.7·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(17.4415\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5464015786\)
\(L(\frac12)\) \(\approx\) \(0.5464015786\)
\(L(2)\) 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)$
bad3$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
good2$C_2^3$ \( 1 - 7 T^{4} + p^{8} T^{8} \)
7$C_2^3$ \( 1 - 4702 T^{4} + p^{8} T^{8} \)
11$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{4} \)
13$C_2^3$ \( 1 - 4222 T^{4} + p^{8} T^{8} \)
17$C_2^3$ \( 1 + 113858 T^{4} + p^{8} T^{8} \)
19$C_2^2$ \( ( 1 - 622 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 475582 T^{4} + p^{8} T^{8} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
31$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )^{4} \)
37$C_2^3$ \( 1 + 3168578 T^{4} + p^{8} T^{8} \)
41$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{4} \)
43$C_2^3$ \( 1 - 2956702 T^{4} + p^{8} T^{8} \)
47$C_2^3$ \( 1 - 9478462 T^{4} + p^{8} T^{8} \)
53$C_2^3$ \( 1 + 15243938 T^{4} + p^{8} T^{8} \)
59$C_2^2$ \( ( 1 + 1138 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{4} \)
67$C_2^3$ \( 1 - 14394142 T^{4} + p^{8} T^{8} \)
71$C_2$ \( ( 1 - 72 T + p^{2} T^{2} )^{4} \)
73$C_2^3$ \( 1 - 10963582 T^{4} + p^{8} T^{8} \)
79$C_2^2$ \( ( 1 - 7582 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 31395742 T^{4} + p^{8} T^{8} \)
89$C_2^2$ \( ( 1 - 7742 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^3$ \( 1 - 171536062 T^{4} + p^{8} T^{8} \)
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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

−10.43272677271919347671140114634, −10.32854973849335610960309268242, −10.08845407323857217880959204922, −9.739890412006283037553083325122, −9.737129714682633257585266768364, −8.941822631590589330935127162669, −8.297992431837071455606485224541, −8.242542159109238893010863315701, −8.196381258044041905517853924566, −7.84828196928130944765311894468, −7.66229350958728451736624291298, −7.27028704252744003276067801359, −6.64449596392523289051848591996, −6.54992475316093479185306346998, −5.71175058250511466770320981869, −5.53765216867931320478999536059, −5.12847651918852031711671823561, −5.11961660639705437874096027671, −4.76273541405197502558615016847, −4.10780310830882473457413560765, −3.10813538946266115431392591894, −3.00536279826085397227320469675, −2.47970882258389291612008452390, −2.26277832427149618763724769302, −0.40958377309226192183101428220, 0.40958377309226192183101428220, 2.26277832427149618763724769302, 2.47970882258389291612008452390, 3.00536279826085397227320469675, 3.10813538946266115431392591894, 4.10780310830882473457413560765, 4.76273541405197502558615016847, 5.11961660639705437874096027671, 5.12847651918852031711671823561, 5.53765216867931320478999536059, 5.71175058250511466770320981869, 6.54992475316093479185306346998, 6.64449596392523289051848591996, 7.27028704252744003276067801359, 7.66229350958728451736624291298, 7.84828196928130944765311894468, 8.196381258044041905517853924566, 8.242542159109238893010863315701, 8.297992431837071455606485224541, 8.941822631590589330935127162669, 9.737129714682633257585266768364, 9.739890412006283037553083325122, 10.08845407323857217880959204922, 10.32854973849335610960309268242, 10.43272677271919347671140114634

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