Properties

Label 8-525e4-1.1-c1e4-0-8
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 − 12·11-s − 4·16-s − 12·19-s + 16·29-s − 2·36-s − 12·44-s − 2·49-s + 28·59-s + 24·61-s − 5·64-s − 12·71-s − 12·76-s + 16·79-s + 3·81-s + 12·89-s + 24·99-s + 52·101-s + 20·109-s + 16·116-s + 46·121-s + 127-s + 131-s + 137-s + 139-s + 8·144-s + ⋯
L(s)  = 1  + 1/2·4-s − 2/3·9-s − 3.61·11-s − 16-s − 2.75·19-s + 2.97·29-s − 1/3·36-s − 1.80·44-s − 2/7·49-s + 3.64·59-s + 3.07·61-s − 5/8·64-s − 1.42·71-s − 1.37·76-s + 1.80·79-s + 1/3·81-s + 1.27·89-s + 2.41·99-s + 5.17·101-s + 1.91·109-s + 1.48·116-s + 4.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2/3·144-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

\(L(1)\) \(\approx\) \(1.150669676\)
\(L(\frac12)\) \(\approx\) \(1.150669676\)
\(L(\frac{3}{2})\) 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$D_4\times C_2$ \( 1 - T^{2} + 5 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
13$D_4\times C_2$ \( 1 - 24 T^{2} + 430 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 40 T^{2} + 926 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 14 T^{2} + 899 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 8 T + 61 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 26 T^{2} + 1035 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$D_4\times C_2$ \( 1 - 74 T^{2} + 3195 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 64 T^{2} + 2894 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 76 T^{2} + 3734 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_4$ \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 46 T^{2} + 2019 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
73$D_4\times C_2$ \( 1 - 248 T^{2} + 25566 T^{4} - 248 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 196 T^{2} + 20054 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 156 T^{2} + 11590 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.85707492403419472580481538524, −7.66528625962246294802361989534, −7.57894015741015216091566339209, −6.84636163051939418062356948804, −6.81718250981032689859539262601, −6.71601904965031689346119385666, −6.61766432208887194782366198082, −6.01959799402150508290441423897, −5.87959130281943471146517372703, −5.72733955661966399703542866562, −5.25926885806113168100867426375, −5.09070978846423539776304633964, −4.86006180053234722446616399071, −4.69078669539520149296832886637, −4.38237457266002397552733504490, −4.03275359468556609396536446866, −3.65662647170474264635971658041, −3.19551621889608181635231193347, −3.01035985076929984123651179510, −2.49657904822927266617452528311, −2.34238718965055737591782505614, −2.15329988216548269930776103203, −2.12758258962844642651920096441, −0.817349675956194043528202335645, −0.42554655813471938205059797383, 0.42554655813471938205059797383, 0.817349675956194043528202335645, 2.12758258962844642651920096441, 2.15329988216548269930776103203, 2.34238718965055737591782505614, 2.49657904822927266617452528311, 3.01035985076929984123651179510, 3.19551621889608181635231193347, 3.65662647170474264635971658041, 4.03275359468556609396536446866, 4.38237457266002397552733504490, 4.69078669539520149296832886637, 4.86006180053234722446616399071, 5.09070978846423539776304633964, 5.25926885806113168100867426375, 5.72733955661966399703542866562, 5.87959130281943471146517372703, 6.01959799402150508290441423897, 6.61766432208887194782366198082, 6.71601904965031689346119385666, 6.81718250981032689859539262601, 6.84636163051939418062356948804, 7.57894015741015216091566339209, 7.66528625962246294802361989534, 7.85707492403419472580481538524

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