Properties

Label 8-75e4-1.1-c8e4-0-1
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $871440.$
Root an. cond. $5.52751$
Motivic weight $8$
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 + 5.02e3·9-s − 1.30e5·16-s − 7.57e4·19-s − 1.40e6·31-s − 8.03e4·36-s + 1.69e7·49-s + 3.01e6·61-s + 3.14e6·64-s + 1.21e6·76-s + 9.19e7·79-s − 1.78e7·81-s + 4.39e8·109-s + 7.60e8·121-s + 2.24e7·124-s + 127-s + 131-s + 137-s + 139-s − 6.57e8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.93e9·169-s − 3.80e8·171-s + ⋯
L(s)  = 1  − 0.0625·4-s + 0.765·9-s − 1.99·16-s − 0.581·19-s − 1.52·31-s − 0.0478·36-s + 2.93·49-s + 0.217·61-s + 0.187·64-s + 0.0363·76-s + 2.36·79-s − 0.414·81-s + 3.11·109-s + 3.54·121-s + 0.0951·124-s − 1.52·144-s + 2.37·169-s − 0.444·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(871440.\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :4, 4, 4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.835223589\)
\(L(\frac12)\) \(\approx\) \(2.835223589\)
\(L(5)\) 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)$
bad3$C_2^2$ \( 1 - 62 p^{4} T^{2} + p^{16} T^{4} \)
5 \( 1 \)
good2$C_2^2$ \( ( 1 + p^{3} T^{2} + p^{16} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 172798 p^{2} T^{2} + p^{16} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 380283362 T^{2} + p^{16} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 969428542 T^{2} + p^{16} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 8342551298 T^{2} + p^{16} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 18938 T + p^{8} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 64711613182 T^{2} + p^{16} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 788066452322 T^{2} + p^{16} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 11338 p T + p^{8} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 5242279978942 T^{2} + p^{16} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 12452468931842 T^{2} + p^{16} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 10942666732702 T^{2} + p^{16} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 30967680304898 T^{2} + p^{16} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 80936075395298 T^{2} + p^{16} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 105562517046242 T^{2} + p^{16} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 753602 T + p^{8} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 806987493281182 T^{2} + p^{16} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 1001758688017922 T^{2} + p^{16} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 847137984315262 T^{2} + p^{16} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 22980982 T + p^{8} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 2352070843223138 T^{2} + p^{16} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 2600204109557762 T^{2} + p^{16} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 6013883197666178 T^{2} + p^{16} T^{4} )^{2} \)
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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

−9.074240173642126724089996978418, −8.705512163853891206415824202640, −8.627534557001692721348142002575, −8.192975350988981454903177889740, −7.71983222506632341171983137917, −7.41730225025252755695091241705, −7.07335665461966739451387103134, −7.03147092777359259468538312687, −6.58170739323311686493351207997, −6.25971987182531883830799790650, −5.80066777165658366929060707671, −5.50905691570727804470144703723, −5.19906899248923815950758829548, −4.51398480959655594650912759312, −4.48655190285847040161065220831, −4.21012150461891235477603698052, −3.68816692226410336731039788634, −3.31156491134432686009725732709, −2.87355011955208419074915529075, −2.14992238683486894247960604533, −1.98675788241854501046629837510, −1.94596074272930092986494057165, −0.972889894753885284454595387830, −0.70112756587193685475036515377, −0.28752246922820041721980186388, 0.28752246922820041721980186388, 0.70112756587193685475036515377, 0.972889894753885284454595387830, 1.94596074272930092986494057165, 1.98675788241854501046629837510, 2.14992238683486894247960604533, 2.87355011955208419074915529075, 3.31156491134432686009725732709, 3.68816692226410336731039788634, 4.21012150461891235477603698052, 4.48655190285847040161065220831, 4.51398480959655594650912759312, 5.19906899248923815950758829548, 5.50905691570727804470144703723, 5.80066777165658366929060707671, 6.25971987182531883830799790650, 6.58170739323311686493351207997, 7.03147092777359259468538312687, 7.07335665461966739451387103134, 7.41730225025252755695091241705, 7.71983222506632341171983137917, 8.192975350988981454903177889740, 8.627534557001692721348142002575, 8.705512163853891206415824202640, 9.074240173642126724089996978418

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