Properties

Label 8-1805e4-1.1-c1e4-0-2
Degree $8$
Conductor $1.061\times 10^{13}$
Sign $1$
Analytic cond. $43153.6$
Root an. cond. $3.79644$
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 + 8·7-s + 8·11-s − 3·16-s + 8·17-s − 8·20-s + 8·23-s + 10·25-s − 16·28-s + 32·35-s − 8·43-s − 16·44-s + 8·47-s + 28·49-s + 32·55-s + 12·64-s − 16·68-s − 24·73-s + 64·77-s − 12·80-s − 10·81-s + 24·83-s + 32·85-s − 16·92-s − 20·100-s − 32·101-s + ⋯
L(s)  = 1  − 4-s + 1.78·5-s + 3.02·7-s + 2.41·11-s − 3/4·16-s + 1.94·17-s − 1.78·20-s + 1.66·23-s + 2·25-s − 3.02·28-s + 5.40·35-s − 1.21·43-s − 2.41·44-s + 1.16·47-s + 4·49-s + 4.31·55-s + 3/2·64-s − 1.94·68-s − 2.80·73-s + 7.29·77-s − 1.34·80-s − 1.11·81-s + 2.63·83-s + 3.47·85-s − 1.66·92-s − 2·100-s − 3.18·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(43153.6\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 19^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(12.81782302\)
\(L(\frac12)\) \(\approx\) \(12.81782302\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
19 \( 1 \)
good2$C_2^2 \wr C_2$ \( 1 + p T^{2} + 7 T^{4} + p^{3} T^{6} + p^{4} T^{8} \)
3$D_4\times C_2$ \( 1 + 10 T^{4} + p^{4} T^{8} \)
7$C_4$ \( ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
13$C_2^2 \wr C_2$ \( 1 + 32 T^{2} + 522 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \)
17$C_4$ \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_4$ \( ( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 + 36 T^{2} + 854 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2^2 \wr C_2$ \( 1 + 76 T^{2} + 3238 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2 \wr C_2$ \( 1 + 96 T^{2} + 4394 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$D_{4}$ \( ( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2 \wr C_2$ \( 1 + 128 T^{2} + 9322 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 + 44 T^{2} + 5398 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2 \wr C_2$ \( 1 + 192 T^{2} + 18122 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 - 20 T^{2} + 9030 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2 \wr C_2$ \( 1 + 124 T^{2} + 14278 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2^2 \wr C_2$ \( 1 + 20 T^{2} + 9670 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2 \wr C_2$ \( 1 + 368 T^{2} + 52602 T^{4} + 368 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.58899569083391258859789035322, −6.56525250184215084237542651339, −6.00256137811455663746926578109, −5.89697178596742466041060977826, −5.71977577739284548533218336007, −5.44551906650689574645642644387, −5.18529548245026991869809452313, −5.17133666843972744940661375285, −4.96204739856865772775916960489, −4.62856822232391527165497431718, −4.51986435273402998445167634174, −4.25738903295857105579407428790, −4.06693877657075806410018931849, −3.93208230217212706617299269682, −3.36398341508515669174902282783, −3.32461300071538811162404986123, −2.83374550228748721832811747765, −2.76780591179750605253961810722, −2.23242539673328336576102811577, −1.89172705282043803470197557471, −1.80536949849057093925698171919, −1.39210179654541350437051642779, −1.23339406749050078765882619428, −1.14612242046054120534481519678, −0.60122646125341031000255316467, 0.60122646125341031000255316467, 1.14612242046054120534481519678, 1.23339406749050078765882619428, 1.39210179654541350437051642779, 1.80536949849057093925698171919, 1.89172705282043803470197557471, 2.23242539673328336576102811577, 2.76780591179750605253961810722, 2.83374550228748721832811747765, 3.32461300071538811162404986123, 3.36398341508515669174902282783, 3.93208230217212706617299269682, 4.06693877657075806410018931849, 4.25738903295857105579407428790, 4.51986435273402998445167634174, 4.62856822232391527165497431718, 4.96204739856865772775916960489, 5.17133666843972744940661375285, 5.18529548245026991869809452313, 5.44551906650689574645642644387, 5.71977577739284548533218336007, 5.89697178596742466041060977826, 6.00256137811455663746926578109, 6.56525250184215084237542651339, 6.58899569083391258859789035322

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