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Properties

Label 8-395e4-1.1-c2e4-0-2
Degree $8$
Conductor $24343800625$
Sign $1$
Analytic cond. $13419.2$
Root an. cond. $3.28069$
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  + 16·4-s + 20·5-s + 160·16-s + 320·20-s + 250·25-s − 68·31-s + 1.28e3·64-s − 316·79-s + 3.20e3·80-s + 83·81-s + 4.00e3·100-s − 772·101-s + 326·121-s − 1.08e3·124-s + 2.50e3·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.36e3·155-s + 157-s + 163-s + 167-s + 676·169-s + 173-s + ⋯
L(s)  = 1  + 4·4-s + 4·5-s + 10·16-s + 16·20-s + 10·25-s − 2.19·31-s + 20·64-s − 4·79-s + 40·80-s + 1.02·81-s + 40·100-s − 7.64·101-s + 2.69·121-s − 8.77·124-s + 20·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 8.77·155-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4·169-s + 0.00578·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 79^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 79^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 79^{4}\)
Sign: $1$
Analytic conductor: \(13419.2\)
Root analytic conductor: \(3.28069\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 79^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(41.35910263\)
\(L(\frac12)\) \(\approx\) \(41.35910263\)
\(L(2)\) 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)$
bad5$C_1$ \( ( 1 - p T )^{4} \)
79$C_1$ \( ( 1 + p T )^{4} \)
good2$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
3$D_4\times C_2$ \( 1 - 83 T^{4} + p^{8} T^{8} \)
7$D_4\times C_2$ \( 1 + 4757 T^{4} + p^{8} T^{8} \)
11$C_2^2$ \( ( 1 - 163 T^{2} + p^{4} T^{4} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
17$D_4\times C_2$ \( 1 - 165778 T^{4} + p^{8} T^{8} \)
19$C_2^2$ \( ( 1 + 542 T^{2} + p^{4} T^{4} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
31$C_2$ \( ( 1 + 17 T + p^{2} T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 + 394517 T^{4} + p^{8} T^{8} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
43$D_4\times C_2$ \( 1 - 2437618 T^{4} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 + 8746862 T^{4} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 7741843 T^{4} + p^{8} T^{8} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
89$C_2^2$ \( ( 1 - 14578 T^{2} + p^{4} T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
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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

−7.86336742206967313290425157129, −7.36112905721647241986403531832, −7.21512647352055497985338439262, −7.09394466369425697914859823258, −6.95490117641597011474818137073, −6.65646861363713681272442734200, −6.37171058167635076705338399355, −6.27067893314366915764828808436, −5.96695645956328703002978792751, −5.74168251373042227735243937321, −5.47640435396063387413609165025, −5.46644285012766727709329092267, −5.33831546053264522313399212959, −4.73200108274403313099754558032, −4.24810604382139638201916734053, −3.59810412575739391208223587727, −3.33944871726197630973056706253, −2.85229373108716556309106362339, −2.83129439724782971388065726953, −2.37344847025815245618818957450, −2.31692878639636401342297462016, −1.90202728408137297071968883065, −1.47105467928402740388675975887, −1.46741718378269457868470324092, −1.13986326295767458377217899329, 1.13986326295767458377217899329, 1.46741718378269457868470324092, 1.47105467928402740388675975887, 1.90202728408137297071968883065, 2.31692878639636401342297462016, 2.37344847025815245618818957450, 2.83129439724782971388065726953, 2.85229373108716556309106362339, 3.33944871726197630973056706253, 3.59810412575739391208223587727, 4.24810604382139638201916734053, 4.73200108274403313099754558032, 5.33831546053264522313399212959, 5.46644285012766727709329092267, 5.47640435396063387413609165025, 5.74168251373042227735243937321, 5.96695645956328703002978792751, 6.27067893314366915764828808436, 6.37171058167635076705338399355, 6.65646861363713681272442734200, 6.95490117641597011474818137073, 7.09394466369425697914859823258, 7.21512647352055497985338439262, 7.36112905721647241986403531832, 7.86336742206967313290425157129

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