Properties

Label 8-21e8-1.1-c1e4-0-5
Degree $8$
Conductor $37822859361$
Sign $1$
Analytic cond. $153.766$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 8·13-s + 4·16-s + 8·19-s − 2·25-s + 8·31-s − 4·37-s − 16·43-s + 8·52-s + 20·61-s + 11·64-s + 8·67-s − 28·73-s + 8·76-s − 16·79-s + 56·97-s − 2·100-s + 8·103-s − 4·109-s + 10·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + ⋯
L(s)  = 1  + 1/2·4-s + 2.21·13-s + 16-s + 1.83·19-s − 2/5·25-s + 1.43·31-s − 0.657·37-s − 2.43·43-s + 1.10·52-s + 2.56·61-s + 11/8·64-s + 0.977·67-s − 3.27·73-s + 0.917·76-s − 1.80·79-s + 5.68·97-s − 1/5·100-s + 0.788·103-s − 0.383·109-s + 0.909·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.328·148-s + 0.0819·149-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: $1$
Analytic conductor: \(153.766\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
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

\(L(1)\) \(\approx\) \(3.751996532\)
\(L(\frac12)\) \(\approx\) \(3.751996532\)
\(L(\frac{3}{2})\) 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$C_2^3$ \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
5$C_2^3$ \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^3$ \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
17$C_2^3$ \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
47$C_2^3$ \( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 58 T^{2} + 555 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 70 T^{2} + 1419 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2^3$ \( 1 - 166 T^{2} + 19635 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
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

−8.246263152268771083765315648130, −7.53241019937762652152513424655, −7.50958530219792359034937777842, −7.48511243474022046708262547843, −7.28216673070781945411917891372, −6.57876256347871427496705803701, −6.45518653816995207796827242672, −6.40141010493233504366606581153, −6.26517661928882596376424624397, −5.61293143883103867370667363314, −5.59467185202947558000675716199, −5.45822745168916558583531875188, −4.96529501032693138863062532402, −4.73923974954126445975856877768, −4.52610011567401792800797087665, −3.76463270586229244986735471166, −3.76179095000196081587617395001, −3.62766619474475797541634169532, −3.22891081079250314690446454487, −2.85636413025418105774924826161, −2.64925095498153368402024617772, −1.84621836664143887549353464714, −1.74670460510648390999590675763, −1.12507189690769241095598355926, −0.828840513664867935599289602033, 0.828840513664867935599289602033, 1.12507189690769241095598355926, 1.74670460510648390999590675763, 1.84621836664143887549353464714, 2.64925095498153368402024617772, 2.85636413025418105774924826161, 3.22891081079250314690446454487, 3.62766619474475797541634169532, 3.76179095000196081587617395001, 3.76463270586229244986735471166, 4.52610011567401792800797087665, 4.73923974954126445975856877768, 4.96529501032693138863062532402, 5.45822745168916558583531875188, 5.59467185202947558000675716199, 5.61293143883103867370667363314, 6.26517661928882596376424624397, 6.40141010493233504366606581153, 6.45518653816995207796827242672, 6.57876256347871427496705803701, 7.28216673070781945411917891372, 7.48511243474022046708262547843, 7.50958530219792359034937777842, 7.53241019937762652152513424655, 8.246263152268771083765315648130

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