Properties

Label 8-1200e4-1.1-c2e4-0-3
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $1.14304\times 10^{6}$
Root an. cond. $5.71818$
Motivic weight $2$
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  + 72·11-s − 88·31-s − 72·41-s + 8·61-s − 288·71-s − 9·81-s − 432·101-s + 2.75e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 6.54·11-s − 2.83·31-s − 1.75·41-s + 8/61·61-s − 4.05·71-s − 1/9·81-s − 4.27·101-s + 22.7·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.14304\times 10^{6}\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3034918279\)
\(L(\frac12)\) \(\approx\) \(0.3034918279\)
\(L(2)\) 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 \( 1 \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
good7$C_2^3$ \( 1 - 4702 T^{4} + p^{8} T^{8} \)
11$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{4} \)
13$C_2^3$ \( 1 - 4222 T^{4} + p^{8} T^{8} \)
17$C_2^3$ \( 1 + 113858 T^{4} + p^{8} T^{8} \)
19$C_2^2$ \( ( 1 - 622 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 475582 T^{4} + p^{8} T^{8} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
31$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{4} \)
37$C_2^3$ \( 1 + 3168578 T^{4} + p^{8} T^{8} \)
41$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{4} \)
43$C_2^3$ \( 1 - 2956702 T^{4} + p^{8} T^{8} \)
47$C_2^3$ \( 1 - 9478462 T^{4} + p^{8} T^{8} \)
53$C_2^3$ \( 1 + 15243938 T^{4} + p^{8} T^{8} \)
59$C_2^2$ \( ( 1 + 1138 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{4} \)
67$C_2^3$ \( 1 - 14394142 T^{4} + p^{8} T^{8} \)
71$C_2$ \( ( 1 + 72 T + p^{2} T^{2} )^{4} \)
73$C_2^3$ \( 1 - 10963582 T^{4} + p^{8} T^{8} \)
79$C_2^2$ \( ( 1 - 7582 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 31395742 T^{4} + p^{8} T^{8} \)
89$C_2^2$ \( ( 1 - 7742 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^3$ \( 1 - 171536062 T^{4} + p^{8} T^{8} \)
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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

−6.82338359782328825452409851045, −6.59263167882699103693159231967, −6.33981180793681909762121562040, −6.06084384357623755753635283428, −5.96263808819646318038187801723, −5.87107162135130488514114328935, −5.61670696789413310450753388140, −5.01060026893579634983812315356, −5.00982737897083204397340122197, −4.63935328473384988718734047012, −4.25614257143678954781283947515, −4.16136351275152470838596548232, −4.00359723737504247754444617890, −3.70086150080348084497032238001, −3.66029235618256847186198273711, −3.39638464842470686883083864571, −3.11196762625280652069811227666, −2.74821410303708808044934980317, −2.02672502410707087109300751169, −2.02453965706723592452500421150, −1.46954510523369084908743948981, −1.38793786638847632015330411662, −1.28014291094949043303090298669, −0.987839078480653736463919096631, −0.06137424429916819463613429294, 0.06137424429916819463613429294, 0.987839078480653736463919096631, 1.28014291094949043303090298669, 1.38793786638847632015330411662, 1.46954510523369084908743948981, 2.02453965706723592452500421150, 2.02672502410707087109300751169, 2.74821410303708808044934980317, 3.11196762625280652069811227666, 3.39638464842470686883083864571, 3.66029235618256847186198273711, 3.70086150080348084497032238001, 4.00359723737504247754444617890, 4.16136351275152470838596548232, 4.25614257143678954781283947515, 4.63935328473384988718734047012, 5.00982737897083204397340122197, 5.01060026893579634983812315356, 5.61670696789413310450753388140, 5.87107162135130488514114328935, 5.96263808819646318038187801723, 6.06084384357623755753635283428, 6.33981180793681909762121562040, 6.59263167882699103693159231967, 6.82338359782328825452409851045

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