Properties

Degree $8$
Conductor $37822859361$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 4·5-s − 2·8-s − 8·10-s − 4·11-s + 16·13-s + 4·17-s + 12·20-s + 8·22-s − 4·23-s + 12·25-s − 32·26-s + 16·29-s + 8·31-s + 6·32-s − 8·34-s + 8·37-s − 8·40-s − 8·41-s − 12·44-s + 8·46-s − 24·50-s + 48·52-s − 4·53-s − 16·55-s − 32·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.78·5-s − 0.707·8-s − 2.52·10-s − 1.20·11-s + 4.43·13-s + 0.970·17-s + 2.68·20-s + 1.70·22-s − 0.834·23-s + 12/5·25-s − 6.27·26-s + 2.97·29-s + 1.43·31-s + 1.06·32-s − 1.37·34-s + 1.31·37-s − 1.26·40-s − 1.24·41-s − 1.80·44-s + 1.17·46-s − 3.39·50-s + 6.65·52-s − 0.549·53-s − 2.15·55-s − 4.20·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.94269\)
\(L(\frac12)\) \(\approx\) \(2.94269\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$D_4\times C_2$ \( 1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
5$D_4\times C_2$ \( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_{4}$ \( ( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 4 T - 4 T^{2} + 56 T^{3} - 161 T^{4} + 56 p T^{5} - 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^3$ \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 4 T - 2 T^{2} - 112 T^{3} - 573 T^{4} - 112 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 8 T - 6 T^{2} - 64 T^{3} + 1955 T^{4} - 64 p T^{5} - 6 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 8 T - 62 T^{2} + 64 T^{3} + 8619 T^{4} + 64 p T^{5} - 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 16 T + 88 T^{2} + 736 T^{3} + 8887 T^{4} + 736 p T^{5} + 88 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^3$ \( 1 - 102 T^{2} + 5915 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 8 T - 656 T^{3} - 5905 T^{4} - 656 p T^{5} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 512 p T^{5} + 66 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 20 T + 140 T^{2} + 1640 T^{3} + 24079 T^{4} + 1640 p T^{5} + 140 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 8 T + 208 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.390079733707483887601553620923, −7.976255430613453068367264961153, −7.54788014737223865923123237174, −7.48127741693162023665546619960, −7.19930000294305810455507696298, −6.64576961261498520870170667197, −6.36778543852952074396382317637, −6.23243657039725400434555352424, −6.21021805321040083135888490447, −6.03515780160396173200725514791, −5.90939540950007645522784922413, −5.23584235566298110067491687426, −5.13875353840431494069549922739, −4.70112146123794961506625974038, −4.52437959695482811351491481854, −4.14317648491255603552024997444, −3.57959267739840412991542623963, −3.42041783992903870632109576289, −2.89347429211605981768832977638, −2.82041242102347821473887324264, −2.46409907578077639279917822344, −1.80929089991354634401080511054, −1.39935063939358553611961969645, −1.20179338033005915762466423335, −0.988486219388395003426174901067, 0.988486219388395003426174901067, 1.20179338033005915762466423335, 1.39935063939358553611961969645, 1.80929089991354634401080511054, 2.46409907578077639279917822344, 2.82041242102347821473887324264, 2.89347429211605981768832977638, 3.42041783992903870632109576289, 3.57959267739840412991542623963, 4.14317648491255603552024997444, 4.52437959695482811351491481854, 4.70112146123794961506625974038, 5.13875353840431494069549922739, 5.23584235566298110067491687426, 5.90939540950007645522784922413, 6.03515780160396173200725514791, 6.21021805321040083135888490447, 6.23243657039725400434555352424, 6.36778543852952074396382317637, 6.64576961261498520870170667197, 7.19930000294305810455507696298, 7.48127741693162023665546619960, 7.54788014737223865923123237174, 7.976255430613453068367264961153, 8.390079733707483887601553620923

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