Properties

Label 8-21e8-1.1-c1e4-0-0
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

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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

\(L(1)\) \(\approx\) \(0.4168885035\)
\(L(\frac12)\) \(\approx\) \(0.4168885035\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 14 T + p T^{2} )^{4} \)
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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

−8.032540759862243909812779302945, −8.026184959205837956307271114123, −7.30784016852034462052284285322, −7.20646216331169981442250147870, −7.15493606615762902651882748294, −7.04237826403988946377928014974, −6.46813622227801216237902914166, −6.45490217576550286973194696935, −6.11017428410712228794791537419, −5.70767997243788889678556460443, −5.53650651591020558543008435591, −5.32031628841393652010864535379, −4.89009565842659254257570699128, −4.76788602342978370105735376727, −4.55036729400383037111854860421, −4.05027739523580402681731346377, −3.76222455969274628938816775019, −3.57164751626833891532980491553, −3.13026816316691568427336711101, −2.78397193683996277240462990076, −2.39308240494693348068871152938, −2.17410195544381373307158306863, −1.70010740464909388783429780781, −1.42585812511258605735415408409, −0.21389328570640586547052829572, 0.21389328570640586547052829572, 1.42585812511258605735415408409, 1.70010740464909388783429780781, 2.17410195544381373307158306863, 2.39308240494693348068871152938, 2.78397193683996277240462990076, 3.13026816316691568427336711101, 3.57164751626833891532980491553, 3.76222455969274628938816775019, 4.05027739523580402681731346377, 4.55036729400383037111854860421, 4.76788602342978370105735376727, 4.89009565842659254257570699128, 5.32031628841393652010864535379, 5.53650651591020558543008435591, 5.70767997243788889678556460443, 6.11017428410712228794791537419, 6.45490217576550286973194696935, 6.46813622227801216237902914166, 7.04237826403988946377928014974, 7.15493606615762902651882748294, 7.20646216331169981442250147870, 7.30784016852034462052284285322, 8.026184959205837956307271114123, 8.032540759862243909812779302945

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