Properties

Degree 8
Conductor $ 2^{4} \cdot 5^{4} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·5-s + 3·9-s − 4·19-s − 2·20-s + 5·25-s + 4·29-s − 20·31-s + 3·36-s − 12·41-s − 6·45-s − 11·49-s + 4·59-s + 18·61-s − 64-s + 24·71-s − 4·76-s − 20·79-s + 9·81-s − 14·89-s + 8·95-s + 5·100-s + 30·101-s + 10·109-s + 4·116-s + 22·121-s − 20·124-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s + 9-s − 0.917·19-s − 0.447·20-s + 25-s + 0.742·29-s − 3.59·31-s + 1/2·36-s − 1.87·41-s − 0.894·45-s − 1.57·49-s + 0.520·59-s + 2.30·61-s − 1/8·64-s + 2.84·71-s − 0.458·76-s − 2.25·79-s + 81-s − 1.48·89-s + 0.820·95-s + 1/2·100-s + 2.98·101-s + 0.957·109-s + 0.371·116-s + 2·121-s − 1.79·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(24010000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{70} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 24010000,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.690768$
$L(\frac12)$  $\approx$  $0.690768$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \)
11$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^3$ \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.04772205434518895150237933479, −10.70567023931131578997706477842, −10.39247182324690179314206702678, −10.13514801880762143898798322353, −9.806660943830376443911478166377, −9.592427995643476242217597018235, −9.011912729514471905709771583413, −8.745418404231952740762255374242, −8.630824227271925815828165466574, −8.047881414401046045353644578488, −7.939813405285696141072269179497, −7.42171420125352958769405301091, −7.12503350874377073437162780492, −6.84056006356308735358610205757, −6.71260132864708786293101035561, −6.21924130930249366732776460977, −5.67341944065894258133949738608, −5.22625661068966867787152024155, −4.94425525517896381904069506670, −4.48479960410171582894100632999, −3.81770243724999369473839296429, −3.72196179733765636182665178792, −3.19474381889672672190380026752, −2.27287846813021968208195845828, −1.72010951584794250908056878961, 1.72010951584794250908056878961, 2.27287846813021968208195845828, 3.19474381889672672190380026752, 3.72196179733765636182665178792, 3.81770243724999369473839296429, 4.48479960410171582894100632999, 4.94425525517896381904069506670, 5.22625661068966867787152024155, 5.67341944065894258133949738608, 6.21924130930249366732776460977, 6.71260132864708786293101035561, 6.84056006356308735358610205757, 7.12503350874377073437162780492, 7.42171420125352958769405301091, 7.939813405285696141072269179497, 8.047881414401046045353644578488, 8.630824227271925815828165466574, 8.745418404231952740762255374242, 9.011912729514471905709771583413, 9.592427995643476242217597018235, 9.806660943830376443911478166377, 10.13514801880762143898798322353, 10.39247182324690179314206702678, 10.70567023931131578997706477842, 11.04772205434518895150237933479

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