Properties

Label 8-7e8-1.1-c5e4-0-4
Degree $8$
Conductor $5764801$
Sign $1$
Analytic cond. $3814.40$
Root an. cond. $2.80335$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·2-s − 9·4-s + 440·8-s − 376·9-s − 1.95e3·11-s − 1.43e3·16-s + 3.76e3·18-s + 1.95e4·22-s − 7.13e3·23-s − 4.86e3·25-s − 3.35e3·29-s + 3.45e3·32-s + 3.38e3·36-s − 9.20e3·37-s + 2.04e4·43-s + 1.75e4·44-s + 7.13e4·46-s + 4.86e4·50-s − 1.02e5·53-s + 3.35e4·58-s − 3.02e4·64-s − 2.28e4·67-s − 1.53e5·71-s − 1.65e5·72-s + 9.20e4·74-s − 9.06e4·79-s + 5.30e4·81-s + ⋯
L(s)  = 1  − 1.76·2-s − 0.281·4-s + 2.43·8-s − 1.54·9-s − 4.86·11-s − 1.39·16-s + 2.73·18-s + 8.59·22-s − 2.81·23-s − 1.55·25-s − 0.740·29-s + 0.595·32-s + 0.435·36-s − 1.10·37-s + 1.68·43-s + 1.36·44-s + 4.97·46-s + 2.75·50-s − 5.03·53-s + 1.30·58-s − 0.922·64-s − 0.623·67-s − 3.62·71-s − 3.76·72-s + 1.95·74-s − 1.63·79-s + 0.897·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5764801 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5764801 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5764801\)    =    \(7^{8}\)
Sign: $1$
Analytic conductor: \(3814.40\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{49} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5764801,\ (\ :5/2, 5/2, 5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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)$
bad7 \( 1 \)
good2$D_{4}$ \( ( 1 + 5 T + 21 p T^{2} + 5 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
3$D_4\times C_2$ \( 1 + 376 T^{2} + 88354 T^{4} + 376 p^{10} T^{6} + p^{20} T^{8} \)
5$D_4\times C_2$ \( 1 + 4868 T^{2} + 22562806 T^{4} + 4868 p^{10} T^{6} + p^{20} T^{8} \)
11$D_{4}$ \( ( 1 + 976 T + 556178 T^{2} + 976 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 224500 T^{2} - 32848089674 T^{4} + 224500 p^{10} T^{6} + p^{20} T^{8} \)
17$D_4\times C_2$ \( 1 + 3065760 T^{2} + 6374095942466 T^{4} + 3065760 p^{10} T^{6} + p^{20} T^{8} \)
19$D_4\times C_2$ \( 1 + 8544504 T^{2} + 30052000046306 T^{4} + 8544504 p^{10} T^{6} + p^{20} T^{8} \)
23$D_{4}$ \( ( 1 + 3568 T + 12712350 T^{2} + 3568 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 1676 T + 24360510 T^{2} + 1676 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 29395756 T^{2} + 658228795954534 T^{4} + 29395756 p^{10} T^{6} + p^{20} T^{8} \)
37$D_{4}$ \( ( 1 + 4604 T + 143922030 T^{2} + 4604 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 212331360 T^{2} + 36724946155085474 T^{4} + 212331360 p^{10} T^{6} + p^{20} T^{8} \)
43$D_{4}$ \( ( 1 - 10224 T + 293018130 T^{2} - 10224 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 568199404 T^{2} + 172542701189040294 T^{4} + 568199404 p^{10} T^{6} + p^{20} T^{8} \)
53$D_{4}$ \( ( 1 + 51460 T + 1308627278 T^{2} + 51460 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 1609944152 T^{2} + 1391734325057519170 T^{4} + 1609944152 p^{10} T^{6} + p^{20} T^{8} \)
61$D_4\times C_2$ \( 1 + 1094138468 T^{2} + 1609346124964468758 T^{4} + 1094138468 p^{10} T^{6} + p^{20} T^{8} \)
67$D_{4}$ \( ( 1 + 11448 T - 1089307338 T^{2} + 11448 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 76912 T + 4562060670 T^{2} + 76912 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 2170565248 T^{2} + 425171557315203874 T^{4} + 2170565248 p^{10} T^{6} + p^{20} T^{8} \)
79$D_{4}$ \( ( 1 + 45344 T + 6154499470 T^{2} + 45344 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 6721792472 T^{2} + 21939991980485194594 T^{4} + 6721792472 p^{10} T^{6} + p^{20} T^{8} \)
89$D_4\times C_2$ \( 1 + 14718300768 T^{2} + \)\(11\!\cdots\!58\)\( T^{4} + 14718300768 p^{10} T^{6} + p^{20} T^{8} \)
97$D_4\times C_2$ \( 1 + 22489794400 T^{2} + 25388976518139394 p^{2} T^{4} + 22489794400 p^{10} T^{6} + p^{20} T^{8} \)
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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

−11.14380907966057588826220282486, −10.58241922657192764114216786963, −10.31181225135767849277232328568, −10.31164073035846273470782939292, −10.02812523454651323689647803562, −9.553415052672121990193250752815, −9.364216183554720916865749405664, −8.863474104461205553391392585154, −8.717782316119673337279916098100, −8.183056734645235509753885554998, −7.982685497822485041401056898698, −7.84784597544561604622415050697, −7.73398342494379724287759921253, −7.42870472198819141502041605806, −6.27995176132619572736980747795, −6.03373819024320781084429844665, −5.67681985381235349729688342946, −5.26916358948914359159788520096, −5.07409854871435889407714482433, −4.44770378485442115297806229410, −4.11671663151358339426894880399, −3.09666836325306029808568338199, −2.81596682940432340996798615827, −2.35933728263432701841238890333, −1.65681012605053875878617953284, 0, 0, 0, 0, 1.65681012605053875878617953284, 2.35933728263432701841238890333, 2.81596682940432340996798615827, 3.09666836325306029808568338199, 4.11671663151358339426894880399, 4.44770378485442115297806229410, 5.07409854871435889407714482433, 5.26916358948914359159788520096, 5.67681985381235349729688342946, 6.03373819024320781084429844665, 6.27995176132619572736980747795, 7.42870472198819141502041605806, 7.73398342494379724287759921253, 7.84784597544561604622415050697, 7.982685497822485041401056898698, 8.183056734645235509753885554998, 8.717782316119673337279916098100, 8.863474104461205553391392585154, 9.364216183554720916865749405664, 9.553415052672121990193250752815, 10.02812523454651323689647803562, 10.31164073035846273470782939292, 10.31181225135767849277232328568, 10.58241922657192764114216786963, 11.14380907966057588826220282486

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