Properties

Label 8-3330e4-1.1-c1e4-0-2
Degree $8$
Conductor $1.230\times 10^{14}$
Sign $1$
Analytic cond. $499902.$
Root an. cond. $5.15656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 4·5-s + 4·11-s + 3·16-s − 12·19-s + 8·20-s + 8·25-s + 24·29-s − 8·31-s − 8·41-s − 8·44-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 − 28·89-s + 48·95-s − 16·100-s − 40·101-s + 20·109-s − 48·116-s − 22·121-s + ⋯
L(s)  = 1  − 4-s − 1.78·5-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 − 1.24·41-s − 1.20·44-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 − 2.96·89-s + 4.92·95-s − 8/5·100-s − 3.98·101-s + 1.91·109-s − 4.45·116-s − 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(499902.\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1896007816\)
\(L(\frac12)\) \(\approx\) \(0.1896007816\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−6.25792361813028399500524683586, −5.99856851177696720981700675202, −5.61641591362258221452029938951, −5.57821811622967548794741336412, −5.10802057240394143915859726117, −4.95415067056705156470920132051, −4.94376402494071040059009933899, −4.50375779239056147760778770670, −4.34051711904962288173508102273, −4.24755707299762839956250689626, −4.14392708996031575963305247401, −3.89542233011209835540455135684, −3.85342378059262724694463509577, −3.55464473982663380654018913902, −2.99745699438390375312687849700, −2.99239730542773081295809570070, −2.75285559808734780785391506923, −2.61347598110972916523937832360, −2.32747192002197616159904850415, −1.72082980332480133707778993565, −1.33460834714662650517374359917, −1.26902562934649661445240191808, −1.24339517099720459801646400496, −0.33951141372234200232497631059, −0.14118841554412059344500254854, 0.14118841554412059344500254854, 0.33951141372234200232497631059, 1.24339517099720459801646400496, 1.26902562934649661445240191808, 1.33460834714662650517374359917, 1.72082980332480133707778993565, 2.32747192002197616159904850415, 2.61347598110972916523937832360, 2.75285559808734780785391506923, 2.99239730542773081295809570070, 2.99745699438390375312687849700, 3.55464473982663380654018913902, 3.85342378059262724694463509577, 3.89542233011209835540455135684, 4.14392708996031575963305247401, 4.24755707299762839956250689626, 4.34051711904962288173508102273, 4.50375779239056147760778770670, 4.94376402494071040059009933899, 4.95415067056705156470920132051, 5.10802057240394143915859726117, 5.57821811622967548794741336412, 5.61641591362258221452029938951, 5.99856851177696720981700675202, 6.25792361813028399500524683586

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