Properties

Label 8-48e4-1.1-c9e4-0-0
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $373520.$
Root an. cond. $4.97209$
Motivic weight $9$
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  + 3.75e4·9-s + 4.49e5·13-s − 3.08e4·25-s + 4.82e7·37-s − 3.45e7·49-s + 1.46e7·61-s − 3.07e8·73-s + 1.02e9·81-s − 2.56e9·97-s − 4.64e9·109-s + 1.68e10·117-s − 9.31e9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.37e10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 1.90·9-s + 4.36·13-s − 0.0158·25-s + 4.23·37-s − 0.855·49-s + 0.135·61-s − 1.26·73-s + 2.63·81-s − 2.94·97-s − 3.15·109-s + 8.31·117-s − 3.95·121-s + 7.89·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(373520.\)
Root analytic conductor: \(4.97209\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(9.241496256\)
\(L(\frac12)\) \(\approx\) \(9.241496256\)
\(L(\frac{11}{2})\) 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 \( 1 \)
3$C_2^2$ \( 1 - 1390 p^{3} T^{2} + p^{18} T^{4} \)
good5$C_2^2$ \( ( 1 + 15446 T^{2} + p^{18} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 352318 p^{2} T^{2} + p^{18} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 4659838198 T^{2} + p^{18} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 112318 T + p^{9} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 1103116034 p T^{2} + p^{18} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 394174226234 T^{2} + p^{18} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 319372399150 T^{2} + p^{18} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 23599900411738 T^{2} + p^{18} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 15530473406734 T^{2} + p^{18} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 12067382 T + p^{9} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 420188579441746 T^{2} + p^{18} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 714546899102090 T^{2} + p^{18} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 365480120259934 T^{2} + p^{18} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 5174988833748490 T^{2} + p^{18} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 9590548198267222 T^{2} + p^{18} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 3669518 T + p^{9} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 54322360686522650 T^{2} + p^{18} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 18423999464530318 T^{2} + p^{18} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 76753334 T + p^{9} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 208797908207424914 T^{2} + p^{18} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 1708268973285382 T^{2} + p^{18} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 7353699092054786 p T^{2} + p^{18} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 641848430 T + p^{9} T^{2} )^{4} \)
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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

−9.603231736774106004473559447412, −9.447365977080014083095206903917, −8.942016459842996463482380200273, −8.771450749306421703226973852564, −8.119700636859432450356682302930, −8.014273473312196400478105062031, −7.945602886307608809075024211221, −7.32928057027102146388204689543, −6.68979822773550269518541643003, −6.66211432962634065761278129669, −6.30893019641287704582197197139, −5.83068955092647475377048208304, −5.78715386843267444141808835876, −5.11936223639677858137868233719, −4.46374341839870865659390744819, −4.15162024818392728676269919506, −3.99847791639741748566105380619, −3.71947420009200577412770413594, −3.07176910771153558364007344770, −2.71508311147826182269381005514, −2.00211853988186855379066940654, −1.26608436761507900943332445503, −1.20872166754901489439922121525, −1.19701410354020692612226605030, −0.43588632454498017463984863997, 0.43588632454498017463984863997, 1.19701410354020692612226605030, 1.20872166754901489439922121525, 1.26608436761507900943332445503, 2.00211853988186855379066940654, 2.71508311147826182269381005514, 3.07176910771153558364007344770, 3.71947420009200577412770413594, 3.99847791639741748566105380619, 4.15162024818392728676269919506, 4.46374341839870865659390744819, 5.11936223639677858137868233719, 5.78715386843267444141808835876, 5.83068955092647475377048208304, 6.30893019641287704582197197139, 6.66211432962634065761278129669, 6.68979822773550269518541643003, 7.32928057027102146388204689543, 7.945602886307608809075024211221, 8.014273473312196400478105062031, 8.119700636859432450356682302930, 8.771450749306421703226973852564, 8.942016459842996463482380200273, 9.447365977080014083095206903917, 9.603231736774106004473559447412

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