Properties

Label 8-882e4-1.1-c2e4-0-7
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $333591.$
Root an. cond. $4.90232$
Motivic weight $2$
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 + 12·16-s − 48·25-s − 24·37-s − 272·43-s − 32·64-s − 416·67-s − 80·79-s + 192·100-s + 468·121-s + 127-s + 131-s + 137-s + 139-s + 96·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 380·169-s + 1.08e3·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 4-s + 3/4·16-s − 1.91·25-s − 0.648·37-s − 6.32·43-s − 1/2·64-s − 6.20·67-s − 1.01·79-s + 1.91·100-s + 3.86·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.648·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.24·169-s + 6.32·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(333591.\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5217759107\)
\(L(\frac12)\) \(\approx\) \(0.5217759107\)
\(L(2)\) 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)$
bad2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 + 24 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 234 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 88 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 130 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 742 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1440 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 1330 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 2696 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 68 T + p^{2} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 318 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 3936 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 2226 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 2030 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 104 T + p^{2} T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 5082 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 6958 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 20 T + p^{2} T^{2} )^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
89$C_2^2$ \( ( 1 - 6888 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 6194 T^{2} + p^{4} T^{4} )^{2} \)
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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

−6.89810729838789633614437076043, −6.82329600200919400285136277497, −6.72879173463548010364093930558, −6.41453302085304345891122172098, −6.04057963136690073635759412224, −5.85164225374559524640314830051, −5.66313341273403205750306222062, −5.53185322348073154281643353134, −5.15951976411244748106373266657, −4.80901252622570268956694178078, −4.67546311757176193439419532991, −4.62880457375277762205533627978, −4.22986449759016546900166220730, −3.94592705394470920443596377872, −3.61632056576438744305955260752, −3.35788222086694448406450937812, −3.27215240928678424396207865802, −2.85537934973616689009891257080, −2.69937931496414362329289544349, −1.87729059707322405065372134023, −1.85510960251154594786740767522, −1.44949666130378794019637614118, −1.40581170529869928932968647518, −0.37708142159957593234561943418, −0.21251314587217680263919261291, 0.21251314587217680263919261291, 0.37708142159957593234561943418, 1.40581170529869928932968647518, 1.44949666130378794019637614118, 1.85510960251154594786740767522, 1.87729059707322405065372134023, 2.69937931496414362329289544349, 2.85537934973616689009891257080, 3.27215240928678424396207865802, 3.35788222086694448406450937812, 3.61632056576438744305955260752, 3.94592705394470920443596377872, 4.22986449759016546900166220730, 4.62880457375277762205533627978, 4.67546311757176193439419532991, 4.80901252622570268956694178078, 5.15951976411244748106373266657, 5.53185322348073154281643353134, 5.66313341273403205750306222062, 5.85164225374559524640314830051, 6.04057963136690073635759412224, 6.41453302085304345891122172098, 6.72879173463548010364093930558, 6.82329600200919400285136277497, 6.89810729838789633614437076043

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