Properties

Label 8-1805e4-1.1-c1e4-0-4
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 $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s − 4·5-s + 2·7-s − 2·9-s − 10·11-s + 40·16-s + 2·17-s + 32·20-s + 4·23-s + 10·25-s − 16·28-s − 8·35-s + 16·36-s − 34·43-s + 80·44-s + 8·45-s − 8·47-s − 23·49-s + 40·55-s − 20·61-s − 4·63-s − 160·64-s − 16·68-s − 44·73-s − 20·77-s − 160·80-s + 5·81-s + ⋯
L(s)  = 1  − 4·4-s − 1.78·5-s + 0.755·7-s − 2/3·9-s − 3.01·11-s + 10·16-s + 0.485·17-s + 7.15·20-s + 0.834·23-s + 2·25-s − 3.02·28-s − 1.35·35-s + 8/3·36-s − 5.18·43-s + 12.0·44-s + 1.19·45-s − 1.16·47-s − 3.28·49-s + 5.39·55-s − 2.56·61-s − 0.503·63-s − 20·64-s − 1.94·68-s − 5.14·73-s − 2.27·77-s − 17.8·80-s + 5/9·81-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: induced by $\chi_{1805} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{4} \cdot 19^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{4} \)
3$C_2^2:C_4$ \( 1 + 2 T^{2} - T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2:C_4$ \( 1 + 42 T^{2} + 759 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
23$D_{4}$ \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 + 96 T^{2} + 3966 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2^2:C_4$ \( 1 + 74 T^{2} + 2791 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2:C_4$ \( 1 + 18 T^{2} + 2799 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 - 6 T^{2} + 951 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 + 17 T + 157 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2:C_4$ \( 1 + 27 T^{2} + 1449 T^{4} + 27 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 + 211 T^{2} + 18061 T^{4} + 211 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2:C_4$ \( 1 + 203 T^{2} + 18829 T^{4} + 203 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2:C_4$ \( 1 + 114 T^{2} + 6111 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{4} \)
79$C_2^2:C_4$ \( 1 - 134 T^{2} + 15351 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 8 T + 137 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2^2:C_4$ \( 1 + 111 T^{2} + 15921 T^{4} + 111 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 + 258 T^{2} + 35439 T^{4} + 258 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.24925219821629660566955284599, −6.91591456807867376754611517098, −6.42676314708237240886196993022, −6.30600223120693907706749402635, −6.20446574084068106303835287022, −5.61490689202384684115410104138, −5.49580290401254317588508523811, −5.42571300836847964032064170405, −5.17802306067682585821016884206, −4.83964967599617408688387679193, −4.77783931152790934855537850894, −4.76023683760472086402988622022, −4.72897109025186701158251897407, −4.34384199858207447852723073499, −3.94236001789148049010090535877, −3.77657602642235815302396227066, −3.55586122555851354197136028429, −3.07140620879446014618282527088, −3.04471750615279069127583911746, −3.03075989187154571245610161709, −2.97490844159120497403468506804, −1.97247302592756378320630139447, −1.57392083943964158900590007818, −1.35422923237514510654852541383, −0.988668033490779323044994557794, 0, 0, 0, 0, 0.988668033490779323044994557794, 1.35422923237514510654852541383, 1.57392083943964158900590007818, 1.97247302592756378320630139447, 2.97490844159120497403468506804, 3.03075989187154571245610161709, 3.04471750615279069127583911746, 3.07140620879446014618282527088, 3.55586122555851354197136028429, 3.77657602642235815302396227066, 3.94236001789148049010090535877, 4.34384199858207447852723073499, 4.72897109025186701158251897407, 4.76023683760472086402988622022, 4.77783931152790934855537850894, 4.83964967599617408688387679193, 5.17802306067682585821016884206, 5.42571300836847964032064170405, 5.49580290401254317588508523811, 5.61490689202384684115410104138, 6.20446574084068106303835287022, 6.30600223120693907706749402635, 6.42676314708237240886196993022, 6.91591456807867376754611517098, 7.24925219821629660566955284599

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