Properties

Label 8-525e4-1.1-c1e4-0-7
Degree $8$
Conductor $75969140625$
Sign $1$
Analytic cond. $308.848$
Root an. cond. $2.04747$
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  + 4-s − 2·9-s − 4·11-s + 4·16-s + 4·19-s + 8·29-s − 2·36-s − 32·41-s − 4·44-s − 2·49-s + 12·59-s − 8·61-s + 11·64-s − 20·71-s + 4·76-s − 16·79-s + 3·81-s − 4·89-s + 8·99-s − 44·101-s − 28·109-s + 8·116-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1/2·4-s − 2/3·9-s − 1.20·11-s + 16-s + 0.917·19-s + 1.48·29-s − 1/3·36-s − 4.99·41-s − 0.603·44-s − 2/7·49-s + 1.56·59-s − 1.02·61-s + 11/8·64-s − 2.37·71-s + 0.458·76-s − 1.80·79-s + 1/3·81-s − 0.423·89-s + 0.804·99-s − 4.37·101-s − 2.68·109-s + 0.742·116-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(308.848\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.173044024\)
\(L(\frac12)\) \(\approx\) \(1.173044024\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( ( 1 + T^{2} )^{2} \)
5 \( 1 \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good2$C_2^3$ \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 24 T^{2} + 302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 24 T^{2} + 542 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 4 T + 17 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 106 T^{2} + 5467 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
43$D_4\times C_2$ \( 1 - 90 T^{2} + 5003 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 128 T^{2} + 8014 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_4$ \( ( 1 + 4 T - 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 146 T^{2} + 11427 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 10 T + 147 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 280 T^{2} + 30238 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 8 T + 169 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 164 T^{2} + 15382 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 2 T + 134 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 220 T^{2} + 25798 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.82779121250595269589007569860, −7.70529078176513594536598361853, −7.33474140706265203583111947204, −7.02135852238868284906466194689, −7.01504206603295341141920793164, −6.53698472330534938503928165471, −6.38209003437364867952162510224, −6.37372105154529693582361468244, −5.76125464253528751313768579938, −5.51662501056955730919185238440, −5.30313492292577676170906596520, −5.25222771584494571758207954451, −4.96767850459350227327135990783, −4.75178273847731043647911131995, −4.08066378282328285310519425593, −3.91637536216849347108536082304, −3.83966297596316986770140245596, −2.97834451304383027415082022637, −2.96822422112917751123288128406, −2.95626352174807229827986193188, −2.66502739460181134897732335368, −1.79005906929978794111508077973, −1.68028554892747885178977757722, −1.27679661244757547946785259237, −0.34176444194724140528034704113, 0.34176444194724140528034704113, 1.27679661244757547946785259237, 1.68028554892747885178977757722, 1.79005906929978794111508077973, 2.66502739460181134897732335368, 2.95626352174807229827986193188, 2.96822422112917751123288128406, 2.97834451304383027415082022637, 3.83966297596316986770140245596, 3.91637536216849347108536082304, 4.08066378282328285310519425593, 4.75178273847731043647911131995, 4.96767850459350227327135990783, 5.25222771584494571758207954451, 5.30313492292577676170906596520, 5.51662501056955730919185238440, 5.76125464253528751313768579938, 6.37372105154529693582361468244, 6.38209003437364867952162510224, 6.53698472330534938503928165471, 7.01504206603295341141920793164, 7.02135852238868284906466194689, 7.33474140706265203583111947204, 7.70529078176513594536598361853, 7.82779121250595269589007569860

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