Properties

Label 8-1110e4-1.1-c1e4-0-9
Degree $8$
Conductor $1.518\times 10^{12}$
Sign $1$
Analytic cond. $6171.63$
Root an. cond. $2.97714$
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 + 4·5-s − 2·9-s − 4·11-s + 3·16-s − 12·19-s − 8·20-s + 8·25-s − 24·29-s − 8·31-s + 4·36-s + 8·41-s + 8·44-s − 8·45-s + 14·49-s − 16·55-s − 16·59-s − 40·61-s − 4·64-s + 24·71-s + 24·76-s − 24·79-s + 12·80-s + 3·81-s + 28·89-s − 48·95-s + 8·99-s + ⋯
L(s)  = 1  − 4-s + 1.78·5-s − 2/3·9-s − 1.20·11-s + 3/4·16-s − 2.75·19-s − 1.78·20-s + 8/5·25-s − 4.45·29-s − 1.43·31-s + 2/3·36-s + 1.24·41-s + 1.20·44-s − 1.19·45-s + 2·49-s − 2.15·55-s − 2.08·59-s − 5.12·61-s − 1/2·64-s + 2.84·71-s + 2.75·76-s − 2.70·79-s + 1.34·80-s + 1/3·81-s + 2.96·89-s − 4.92·95-s + 0.804·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(6171.63\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.401009033\)
\(L(\frac12)\) \(\approx\) \(1.401009033\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
good7$D_4\times C_2$ \( 1 - 2 p T^{2} + 123 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + 2 T + 17 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 22 T^{2} + 243 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 54 T^{2} + 1283 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 12 T + 88 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
41$D_{4}$ \( ( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 116 T^{2} + 6678 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 144 T^{2} + 9218 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 + 8 T + 80 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 212 T^{2} + 19830 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 98 T^{2} + 12675 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 12 T + 140 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 318 T^{2} + 39035 T^{4} - 318 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 14 T + 173 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 248 T^{2} + 30738 T^{4} - 248 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.08458014835453609876088330698, −6.80895183879384306910343152432, −6.44613257633141442996200539551, −6.15575314427819058113314936972, −6.12425294997650808160273060812, −5.78863081029412090435757443259, −5.62110540793100754369842636544, −5.60701917596036547908785574005, −5.55041574340890509808309767954, −4.82861739546831656081622888815, −4.78312513816961127053228106504, −4.70851676198374917926048635415, −4.20853763565165879947857924853, −4.14262289777883781673982093007, −3.80956787097617649081673133121, −3.32833039938196138978049009932, −3.28871522073292721679674528602, −2.98402082858061395725993369768, −2.55712830449778924058979814898, −2.12073114387487278879960720074, −1.91030552513461033261332546994, −1.84290282034480781219204791631, −1.70328877473667172111999358134, −0.52247514974945832890410220973, −0.41200896554588044472365426993, 0.41200896554588044472365426993, 0.52247514974945832890410220973, 1.70328877473667172111999358134, 1.84290282034480781219204791631, 1.91030552513461033261332546994, 2.12073114387487278879960720074, 2.55712830449778924058979814898, 2.98402082858061395725993369768, 3.28871522073292721679674528602, 3.32833039938196138978049009932, 3.80956787097617649081673133121, 4.14262289777883781673982093007, 4.20853763565165879947857924853, 4.70851676198374917926048635415, 4.78312513816961127053228106504, 4.82861739546831656081622888815, 5.55041574340890509808309767954, 5.60701917596036547908785574005, 5.62110540793100754369842636544, 5.78863081029412090435757443259, 6.12425294997650808160273060812, 6.15575314427819058113314936972, 6.44613257633141442996200539551, 6.80895183879384306910343152432, 7.08458014835453609876088330698

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