Properties

Label 8-384e4-1.1-c3e4-0-2
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $263505.$
Root an. cond. $4.75990$
Motivic weight $3$
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  + 18·9-s − 576·23-s − 436·25-s − 2.30e3·47-s + 796·49-s − 2.88e3·71-s − 712·73-s − 405·81-s + 2.60e3·97-s + 3.52e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.58e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 2/3·9-s − 5.22·23-s − 3.48·25-s − 7.15·47-s + 2.32·49-s − 4.81·71-s − 1.14·73-s − 5/9·81-s + 2.72·97-s + 2.64·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.719·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(263505.\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.2294941857\)
\(L(\frac12)\) \(\approx\) \(0.2294941857\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 2 p^{2} T^{2} + p^{6} T^{4} \)
good5$C_2^2$ \( ( 1 + 218 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 398 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 1762 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 790 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 7234 T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 13070 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2$ \( ( 1 + 144 T + p^{3} T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 48746 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 10910 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 96122 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 44530 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 106526 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2$ \( ( 1 + 576 T + p^{3} T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 32762 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 239362 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 199946 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 21202 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 720 T + p^{3} T^{2} )^{4} \)
73$C_2$ \( ( 1 + 178 T + p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 50366 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 951730 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 660850 T^{2} + p^{6} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 650 T + p^{3} T^{2} )^{4} \)
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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

−7.80722387709199429916804167159, −7.63657467797307740312754406415, −7.51505926995414771743941616050, −7.07786769427252295565252297856, −6.58994381961976050526281556992, −6.39626650088952568682881263892, −6.32584564579157437845224434222, −5.89012125519471838152103540997, −5.83378996238198591387819113968, −5.61602339527864029087998131985, −5.29933565141488361177774835691, −4.74086984240893309868163592310, −4.43271045994656069647611883641, −4.22363940388934884837467127752, −4.19760928153514065926102985957, −3.77767869185827828662869807685, −3.42407954707002348390106755311, −3.17326869401691877885573219438, −2.78973873002070409550126386262, −1.95402057710093054964865053480, −1.94336330063024409060053066749, −1.73892460011998840207050988012, −1.61469915683566450054172956076, −0.45623467062911906826079450368, −0.11271055687836309254715393360, 0.11271055687836309254715393360, 0.45623467062911906826079450368, 1.61469915683566450054172956076, 1.73892460011998840207050988012, 1.94336330063024409060053066749, 1.95402057710093054964865053480, 2.78973873002070409550126386262, 3.17326869401691877885573219438, 3.42407954707002348390106755311, 3.77767869185827828662869807685, 4.19760928153514065926102985957, 4.22363940388934884837467127752, 4.43271045994656069647611883641, 4.74086984240893309868163592310, 5.29933565141488361177774835691, 5.61602339527864029087998131985, 5.83378996238198591387819113968, 5.89012125519471838152103540997, 6.32584564579157437845224434222, 6.39626650088952568682881263892, 6.58994381961976050526281556992, 7.07786769427252295565252297856, 7.51505926995414771743941616050, 7.63657467797307740312754406415, 7.80722387709199429916804167159

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