Properties

Label 8-80e8-1.1-c1e4-0-10
Degree $8$
Conductor $1.678\times 10^{15}$
Sign $1$
Analytic cond. $6.82069\times 10^{6}$
Root an. cond. $7.14872$
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  − 2·9-s − 12·17-s − 12·41-s − 28·49-s − 4·73-s + 9·81-s − 36·89-s − 40·97-s − 36·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 2/3·9-s − 2.91·17-s − 1.87·41-s − 4·49-s − 0.468·73-s + 81-s − 3.81·89-s − 4.06·97-s − 3.38·113-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(6.82069\times 10^{6}\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{32} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2^3$ \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
7$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$C_2^3$ \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^3$ \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{4} \)
67$C_2^3$ \( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^3$ \( 1 - 158 T^{2} + 18075 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
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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

−6.08316556266648971830728938643, −5.66727363198885327708673664565, −5.66096935531862489548177663763, −5.53863460502976753827261157159, −5.29344322942415186543109693780, −4.96494250297000239524104258498, −4.77430953879237443375719650367, −4.71891998285243366848637964801, −4.67489823635843136447572106622, −4.35729590380855652173087865507, −4.01594161833870883294481919283, −3.93487723900185198218735131371, −3.85275226924941843631452294886, −3.47396918718054160660739750244, −3.23557334039959480290541199084, −3.02092072988580322889793501959, −2.97072875747322522976592248663, −2.60234150203029626329223388867, −2.37525137814252468389312573165, −2.30907265445140981471620614121, −1.90500964987917448452634149478, −1.83755944889717151008099956773, −1.28290468989391920260989659965, −1.27324707778811480902483339128, −1.13692280252454354787850636147, 0, 0, 0, 0, 1.13692280252454354787850636147, 1.27324707778811480902483339128, 1.28290468989391920260989659965, 1.83755944889717151008099956773, 1.90500964987917448452634149478, 2.30907265445140981471620614121, 2.37525137814252468389312573165, 2.60234150203029626329223388867, 2.97072875747322522976592248663, 3.02092072988580322889793501959, 3.23557334039959480290541199084, 3.47396918718054160660739750244, 3.85275226924941843631452294886, 3.93487723900185198218735131371, 4.01594161833870883294481919283, 4.35729590380855652173087865507, 4.67489823635843136447572106622, 4.71891998285243366848637964801, 4.77430953879237443375719650367, 4.96494250297000239524104258498, 5.29344322942415186543109693780, 5.53863460502976753827261157159, 5.66096935531862489548177663763, 5.66727363198885327708673664565, 6.08316556266648971830728938643

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