Properties

Label 8-725e4-1.1-c1e4-0-3
Degree $8$
Conductor $276281640625$
Sign $1$
Analytic cond. $1123.20$
Root an. cond. $2.40606$
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  + 2·4-s + 6·9-s + 4·11-s + 3·16-s − 24·19-s − 4·29-s + 12·31-s + 12·36-s + 16·41-s + 8·44-s + 12·49-s − 8·59-s − 8·61-s + 12·64-s − 24·71-s − 48·76-s + 4·79-s + 17·81-s + 16·89-s + 24·99-s − 32·101-s − 28·109-s − 8·116-s − 30·121-s + 24·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 4-s + 2·9-s + 1.20·11-s + 3/4·16-s − 5.50·19-s − 0.742·29-s + 2.15·31-s + 2·36-s + 2.49·41-s + 1.20·44-s + 12/7·49-s − 1.04·59-s − 1.02·61-s + 3/2·64-s − 2.84·71-s − 5.50·76-s + 0.450·79-s + 17/9·81-s + 1.69·89-s + 2.41·99-s − 3.18·101-s − 2.68·109-s − 0.742·116-s − 2.72·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 29^{4}\)
Sign: $1$
Analytic conductor: \(1123.20\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 29^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.698872079\)
\(L(\frac12)\) \(\approx\) \(3.698872079\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
29$C_1$ \( ( 1 + T )^{4} \)
good2$D_4\times C_2$ \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
3$D_4\times C_2$ \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \)
7$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 34 T^{2} + 595 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
23$D_4\times C_2$ \( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 - 6 T + 21 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 118 T^{2} + 6979 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 150 T^{2} + 9971 T^{4} - 150 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 66 T^{2} + 6419 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
71$C_4$ \( ( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 2 T + 157 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 260 T^{2} + 30166 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 212 T^{2} + 25446 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.47657781401479545718473787648, −7.11944565241574431154064201003, −7.02253790501487684012798192387, −6.66324152571396465483351384904, −6.49100948703680544579247631642, −6.38206756262511560604074940158, −6.25678719609871240587496123779, −6.08437743879278583693283662347, −5.73124350465580810557285279325, −5.39009931274938719295779736031, −5.02060175767515359923943781842, −4.62726593830247360392498981361, −4.35492792402716654153039549420, −4.25924472918317891439284689760, −4.10145923550051239354910802045, −3.87062475494742786256579277328, −3.85610452203517744381777545836, −2.90705743895005669493145436931, −2.81766007168733967353649384265, −2.46924810676940573401832977492, −2.22774949831765736529290230890, −1.70564469692638566916596330688, −1.66131748018236444571218297353, −1.20956556050702346931000931835, −0.49183566482458569762215677730, 0.49183566482458569762215677730, 1.20956556050702346931000931835, 1.66131748018236444571218297353, 1.70564469692638566916596330688, 2.22774949831765736529290230890, 2.46924810676940573401832977492, 2.81766007168733967353649384265, 2.90705743895005669493145436931, 3.85610452203517744381777545836, 3.87062475494742786256579277328, 4.10145923550051239354910802045, 4.25924472918317891439284689760, 4.35492792402716654153039549420, 4.62726593830247360392498981361, 5.02060175767515359923943781842, 5.39009931274938719295779736031, 5.73124350465580810557285279325, 6.08437743879278583693283662347, 6.25678719609871240587496123779, 6.38206756262511560604074940158, 6.49100948703680544579247631642, 6.66324152571396465483351384904, 7.02253790501487684012798192387, 7.11944565241574431154064201003, 7.47657781401479545718473787648

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