Properties

Label 8-825e4-1.1-c1e4-0-10
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $1883.32$
Root an. cond. $2.56664$
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 − 2·9-s − 4·11-s + 3·16-s + 16·19-s + 8·29-s − 4·36-s + 8·41-s − 8·44-s + 4·49-s + 16·59-s − 24·61-s + 12·64-s + 32·71-s + 32·76-s + 3·81-s + 8·89-s + 8·99-s + 8·101-s − 24·109-s + 16·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + ⋯
L(s)  = 1  + 4-s − 2/3·9-s − 1.20·11-s + 3/4·16-s + 3.67·19-s + 1.48·29-s − 2/3·36-s + 1.24·41-s − 1.20·44-s + 4/7·49-s + 2.08·59-s − 3.07·61-s + 3/2·64-s + 3.79·71-s + 3.67·76-s + 1/3·81-s + 0.847·89-s + 0.804·99-s + 0.796·101-s − 2.29·109-s + 1.48·116-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1883.32\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.767295564\)
\(L(\frac12)\) \(\approx\) \(4.767295564\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 + T )^{4} \)
good2$D_4\times C_2$ \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
7$C_4\times C_2$ \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 12 T^{2} - 1834 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 52 T^{2} + 4246 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
61$D_{4}$ \( ( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 252 T^{2} + 30086 T^{4} - 252 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.48775495463780875543976542710, −7.08290136686817891922072277247, −6.95885925193177750885125328827, −6.61953017175350998789262794380, −6.49915623841898382295793606365, −6.25714136758179875364443566652, −5.82353290354193754799186635818, −5.69058803795240891829749169166, −5.51654273147245108412606694146, −5.22276486110793439709076503918, −5.13321722628245346220217454604, −4.90143341949170828001505059695, −4.51050717951824902994725217071, −4.28667896326212264251382151474, −3.63197644658420660360651121604, −3.58912204404934569321063156039, −3.48289237396234181886523872791, −2.86614929481150183736850067960, −2.80180300784318289124310698989, −2.62869469164556863918922539673, −2.30864727305991836298526744164, −1.84308860943031413382342096906, −1.35448624701722751053163654461, −0.885607331966827786438884359069, −0.67388100516465070142553089686, 0.67388100516465070142553089686, 0.885607331966827786438884359069, 1.35448624701722751053163654461, 1.84308860943031413382342096906, 2.30864727305991836298526744164, 2.62869469164556863918922539673, 2.80180300784318289124310698989, 2.86614929481150183736850067960, 3.48289237396234181886523872791, 3.58912204404934569321063156039, 3.63197644658420660360651121604, 4.28667896326212264251382151474, 4.51050717951824902994725217071, 4.90143341949170828001505059695, 5.13321722628245346220217454604, 5.22276486110793439709076503918, 5.51654273147245108412606694146, 5.69058803795240891829749169166, 5.82353290354193754799186635818, 6.25714136758179875364443566652, 6.49915623841898382295793606365, 6.61953017175350998789262794380, 6.95885925193177750885125328827, 7.08290136686817891922072277247, 7.48775495463780875543976542710

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