Properties

Label 8-637e4-1.1-c1e4-0-4
Degree $8$
Conductor $164648481361$
Sign $1$
Analytic cond. $669.369$
Root an. cond. $2.25532$
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  − 4·3-s + 4-s + 4·9-s + 12·11-s − 4·12-s − 4·13-s + 4·16-s − 12·17-s − 8·19-s − 6·23-s + 7·25-s + 4·27-s + 6·29-s − 2·31-s − 48·33-s + 4·36-s + 14·37-s + 16·39-s − 10·43-s + 12·44-s + 12·47-s − 16·48-s + 48·51-s − 4·52-s + 6·53-s + 32·57-s + 18·59-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s + 4/3·9-s + 3.61·11-s − 1.15·12-s − 1.10·13-s + 16-s − 2.91·17-s − 1.83·19-s − 1.25·23-s + 7/5·25-s + 0.769·27-s + 1.11·29-s − 0.359·31-s − 8.35·33-s + 2/3·36-s + 2.30·37-s + 2.56·39-s − 1.52·43-s + 1.80·44-s + 1.75·47-s − 2.30·48-s + 6.72·51-s − 0.554·52-s + 0.824·53-s + 4.23·57-s + 2.34·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(7^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(669.369\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 7^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5753926190\)
\(L(\frac12)\) \(\approx\) \(0.5753926190\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
good2$C_2^3$ \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
3$D_{4}$ \( ( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
5$C_2^3$ \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 12 T + 77 T^{2} + 396 T^{3} + 1752 T^{4} + 396 p T^{5} + 77 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
23$D_4\times C_2$ \( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 2 T - 32 T^{2} - 52 T^{3} + 211 T^{4} - 52 p T^{5} - 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^3$ \( 1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 10 T + 16 T^{2} - 20 T^{3} + 907 T^{4} - 20 p T^{5} + 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 12 T + 62 T^{2} + 144 T^{3} - 2253 T^{4} + 144 p T^{5} + 62 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 6 T - 31 T^{2} + 234 T^{3} - 228 T^{4} + 234 p T^{5} - 31 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 18 T + 128 T^{2} - 1404 T^{3} + 15819 T^{4} - 1404 p T^{5} + 128 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 - 20 T + 195 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 4 T - 107 T^{2} + 92 T^{3} + 8632 T^{4} + 92 p T^{5} - 107 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 22 T + 232 T^{2} + 2068 T^{3} + 19027 T^{4} + 2068 p T^{5} + 232 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 12 T - 22 T^{2} + 144 T^{3} + 6819 T^{4} + 144 p T^{5} - 22 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 8 T - 38 T^{2} - 736 T^{3} - 5213 T^{4} - 736 p T^{5} - 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.41449916991544366332277550046, −7.01799562307966715549023865822, −6.95412719244024001048982664383, −6.68107704362058089525126328432, −6.67175064722154479665452437129, −6.53310914041835502023902267148, −6.21302402540061910385309301122, −6.11617526533298091514399644178, −5.67350714520644949779872395030, −5.52220009879049913764546464597, −5.39533433714218930617399785973, −5.07064162014393794548973120104, −4.49666343947144302044784506850, −4.37522102676400650828390087537, −4.23342338249758058683365163570, −4.12553371873818115213897760704, −3.66016950619840959297246859945, −3.53948702228426859352325265408, −2.60024728442608355251078208551, −2.58382009843748109378909568535, −2.24298650214658793330415730834, −1.97547233017354786417857604930, −1.10538539451882305493523554086, −1.08764854122519380159396951511, −0.31028797259434859721374532406, 0.31028797259434859721374532406, 1.08764854122519380159396951511, 1.10538539451882305493523554086, 1.97547233017354786417857604930, 2.24298650214658793330415730834, 2.58382009843748109378909568535, 2.60024728442608355251078208551, 3.53948702228426859352325265408, 3.66016950619840959297246859945, 4.12553371873818115213897760704, 4.23342338249758058683365163570, 4.37522102676400650828390087537, 4.49666343947144302044784506850, 5.07064162014393794548973120104, 5.39533433714218930617399785973, 5.52220009879049913764546464597, 5.67350714520644949779872395030, 6.11617526533298091514399644178, 6.21302402540061910385309301122, 6.53310914041835502023902267148, 6.67175064722154479665452437129, 6.68107704362058089525126328432, 6.95412719244024001048982664383, 7.01799562307966715549023865822, 7.41449916991544366332277550046

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