Properties

Label 8-80e8-1.1-c1e4-0-0
Degree $8$
Conductor $1.678\times 10^{15}$
Sign $1$
Analytic cond. $6.82069\times 10^{6}$
Root an. cond. $7.14872$
Motivic weight $1$
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  − 12·9-s + 8·41-s − 12·49-s + 90·81-s + 56·89-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 4·9-s + 1.24·41-s − 1.71·49-s + 10·81-s + 5.93·89-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(6.82069\times 10^{6}\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5424214265\)
\(L(\frac12)\) \(\approx\) \(0.5424214265\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )^{4} \)
7$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−5.72690018408399215610948321705, −5.42037356405753131005106594040, −5.22886775505393823359938493183, −5.21616069665935701661215643404, −4.96252745119533619033067413305, −4.68724119004599242752577528260, −4.59507693393422520929209621113, −4.48657101389067231080574766005, −4.07240999460776770777394491068, −3.65733158171779695015431377047, −3.52413751406553569516076847105, −3.50656485651387586380541926823, −3.47993731638028744469805292850, −2.97428307259247394288770954499, −2.83985237177866916130144251848, −2.68938403856704086596122066369, −2.55715243945473302687946450398, −2.29304543133356310196263779196, −1.95705241053648723725728959061, −1.74194202384615726561746861635, −1.71681593148920776866663034263, −0.873541701307198057398745425448, −0.71570579983271851545100075672, −0.66484135115214633738124175840, −0.12212941096879581942730283271, 0.12212941096879581942730283271, 0.66484135115214633738124175840, 0.71570579983271851545100075672, 0.873541701307198057398745425448, 1.71681593148920776866663034263, 1.74194202384615726561746861635, 1.95705241053648723725728959061, 2.29304543133356310196263779196, 2.55715243945473302687946450398, 2.68938403856704086596122066369, 2.83985237177866916130144251848, 2.97428307259247394288770954499, 3.47993731638028744469805292850, 3.50656485651387586380541926823, 3.52413751406553569516076847105, 3.65733158171779695015431377047, 4.07240999460776770777394491068, 4.48657101389067231080574766005, 4.59507693393422520929209621113, 4.68724119004599242752577528260, 4.96252745119533619033067413305, 5.21616069665935701661215643404, 5.22886775505393823359938493183, 5.42037356405753131005106594040, 5.72690018408399215610948321705

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