Properties

Label 8-65e8-1.1-c1e4-0-0
Degree $8$
Conductor $3.186\times 10^{14}$
Sign $1$
Analytic cond. $1.29543\times 10^{6}$
Root an. cond. $5.80833$
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  − 2·2-s + 2·3-s − 4-s − 4·6-s − 10·7-s + 4·8-s − 2·9-s − 2·12-s + 20·14-s + 2·17-s + 4·18-s + 16·19-s − 20·21-s + 10·23-s + 8·24-s − 8·27-s + 10·28-s + 8·29-s + 8·31-s − 2·32-s − 4·34-s + 2·36-s + 2·37-s − 32·38-s + 8·41-s + 40·42-s + 2·43-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s − 1/2·4-s − 1.63·6-s − 3.77·7-s + 1.41·8-s − 2/3·9-s − 0.577·12-s + 5.34·14-s + 0.485·17-s + 0.942·18-s + 3.67·19-s − 4.36·21-s + 2.08·23-s + 1.63·24-s − 1.53·27-s + 1.88·28-s + 1.48·29-s + 1.43·31-s − 0.353·32-s − 0.685·34-s + 1/3·36-s + 0.328·37-s − 5.19·38-s + 1.24·41-s + 6.17·42-s + 0.304·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(1.29543\times 10^{6}\)
Root analytic conductor: \(5.80833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.010595255\)
\(L(\frac12)\) \(\approx\) \(2.010595255\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
13 \( 1 \)
good2$D_4\times C_2$ \( 1 + p T + 5 T^{2} + p^{3} T^{3} + 13 T^{4} + p^{4} T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
3$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 2 p T^{2} - 8 T^{3} + 19 T^{4} - 8 p T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 58 T^{2} + 232 T^{3} + 703 T^{4} + 232 p T^{5} + 58 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 + 14 T^{2} + 9 p T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 2 T + 50 T^{2} - 92 T^{3} + 1135 T^{4} - 92 p T^{5} + 50 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 8 T + 51 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 98 T^{2} - 544 T^{3} + 137 p T^{4} - 544 p T^{5} + 98 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 98 T^{2} - 656 T^{3} + 4003 T^{4} - 656 p T^{5} + 98 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 94 T^{2} - 260 T^{3} + 4219 T^{4} - 260 p T^{5} + 94 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 4 T + 83 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 154 T^{2} - 248 T^{3} + 9559 T^{4} - 248 p T^{5} + 154 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 116 T^{2} + 392 T^{3} + 5158 T^{4} + 392 p T^{5} + 116 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 248 T^{2} - 36 p T^{3} + 20622 T^{4} - 36 p^{2} T^{5} + 248 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 266 T^{2} - 2136 T^{3} + 24423 T^{4} - 2136 p T^{5} + 266 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 28 T + 502 T^{2} - 6088 T^{3} + 55063 T^{4} - 6088 p T^{5} + 502 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 30 T + 598 T^{2} + 7608 T^{3} + 73923 T^{4} + 7608 p T^{5} + 598 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 74 T^{2} - 424 T^{3} + 10903 T^{4} - 424 p T^{5} + 74 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 208 T^{2} - 920 T^{3} + 17998 T^{4} - 920 p T^{5} + 208 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 184 T^{2} + 1256 T^{3} + 21022 T^{4} + 1256 p T^{5} + 184 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 308 T^{2} - 2700 T^{3} + 37158 T^{4} - 2700 p T^{5} + 308 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 122 T^{2} + 1056 T^{3} + 14727 T^{4} + 1056 p T^{5} + 122 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 298 T^{2} + 956 T^{3} + 38551 T^{4} + 956 p T^{5} + 298 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.93954391346405088334786803038, −5.79442231969903663509172193604, −5.65901171085087409110344300567, −5.38464026816075382585926093840, −5.28009847982693619413447572296, −4.77289239589637991497589320890, −4.76909852104274917697264579268, −4.65707378579428407107972453239, −4.31228133988233489712828644923, −3.99021211006077831743406507250, −3.60075717165485199145453344471, −3.53428883437294555433550297231, −3.30871171834445249982431905247, −3.16091299860889454942666357025, −3.15550906216449768702011007655, −2.94938307908908578423705918291, −2.72047213376866170456678279511, −2.55044516287328545730059277893, −2.18541160804132786755557800103, −1.87525859458564414416460482275, −1.26545098273661270542718388480, −0.845913340537089320075118773641, −0.70791127681211863382592782506, −0.61417710783734846700004472851, −0.51913145495065955529476892799, 0.51913145495065955529476892799, 0.61417710783734846700004472851, 0.70791127681211863382592782506, 0.845913340537089320075118773641, 1.26545098273661270542718388480, 1.87525859458564414416460482275, 2.18541160804132786755557800103, 2.55044516287328545730059277893, 2.72047213376866170456678279511, 2.94938307908908578423705918291, 3.15550906216449768702011007655, 3.16091299860889454942666357025, 3.30871171834445249982431905247, 3.53428883437294555433550297231, 3.60075717165485199145453344471, 3.99021211006077831743406507250, 4.31228133988233489712828644923, 4.65707378579428407107972453239, 4.76909852104274917697264579268, 4.77289239589637991497589320890, 5.28009847982693619413447572296, 5.38464026816075382585926093840, 5.65901171085087409110344300567, 5.79442231969903663509172193604, 5.93954391346405088334786803038

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