Properties

Label 8-1792e4-1.1-c2e4-0-5
Degree $8$
Conductor $1.031\times 10^{13}$
Sign $1$
Analytic cond. $5.68449\times 10^{6}$
Root an. cond. $6.98773$
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  + 20·9-s − 8·17-s − 28·25-s − 184·41-s − 14·49-s + 440·73-s + 138·81-s + 536·89-s − 712·97-s − 232·113-s − 260·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 160·153-s + 157-s + 163-s + 167-s + 644·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 20/9·9-s − 0.470·17-s − 1.11·25-s − 4.48·41-s − 2/7·49-s + 6.02·73-s + 1.70·81-s + 6.02·89-s − 7.34·97-s − 2.05·113-s − 2.14·121-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 − 1.04·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.81·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(5.68449\times 10^{6}\)
Root analytic conductor: \(6.98773\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.137562537\)
\(L(\frac12)\) \(\approx\) \(5.137562537\)
\(L(2)\) 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 \( 1 \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{2} \)
5$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{2}( 1 + 6 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( ( 1 + 130 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 322 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 22 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 610 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1486 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 914 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2542 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 3586 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 3410 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 5134 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 1130 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 5138 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 4946 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 114 T + p^{2} T^{2} )^{2}( 1 + 114 T + p^{2} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 110 T + p^{2} T^{2} )^{4} \)
79$C_2$ \( ( 1 - 94 T + p^{2} T^{2} )^{2}( 1 + 94 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 12406 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 134 T + p^{2} T^{2} )^{4} \)
97$C_2$ \( ( 1 + 178 T + p^{2} T^{2} )^{4} \)
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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

−6.57202011086652406896330687983, −6.44542927038635802470132251414, −6.12384983851452613405922468814, −5.66371848822991583414372584449, −5.30386998247458265377416433310, −5.26902022659842603376138170248, −5.26869351383162343475770333213, −4.98898410441577187226332901314, −4.63291803165471728373241243344, −4.46432152008527140063466961512, −4.11266647342398292835385971645, −3.98510105408006794830704173377, −3.82473008076159307553324137617, −3.59471368711522682586302446378, −3.30921427660506797051177444346, −3.13635370477787079124205920776, −2.61890822058894839096763045108, −2.49847903571140027700360859891, −1.99732387324401746489322676100, −1.82688788109700823564274267422, −1.65539907209399692178895231386, −1.48874907856433797115476754129, −0.962374652839563558891479346768, −0.53726040411416835850738460761, −0.33866213828898001164804420824, 0.33866213828898001164804420824, 0.53726040411416835850738460761, 0.962374652839563558891479346768, 1.48874907856433797115476754129, 1.65539907209399692178895231386, 1.82688788109700823564274267422, 1.99732387324401746489322676100, 2.49847903571140027700360859891, 2.61890822058894839096763045108, 3.13635370477787079124205920776, 3.30921427660506797051177444346, 3.59471368711522682586302446378, 3.82473008076159307553324137617, 3.98510105408006794830704173377, 4.11266647342398292835385971645, 4.46432152008527140063466961512, 4.63291803165471728373241243344, 4.98898410441577187226332901314, 5.26869351383162343475770333213, 5.26902022659842603376138170248, 5.30386998247458265377416433310, 5.66371848822991583414372584449, 6.12384983851452613405922468814, 6.44542927038635802470132251414, 6.57202011086652406896330687983

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