Properties

Label 8-322e4-1.1-c1e4-0-1
Degree $8$
Conductor $10750371856$
Sign $1$
Analytic cond. $43.7050$
Root an. cond. $1.60349$
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  − 4·2-s + 10·4-s − 20·8-s + 8·9-s + 35·16-s − 32·18-s − 12·23-s − 20·25-s + 24·29-s − 56·32-s + 80·36-s + 48·46-s + 14·49-s + 80·50-s − 96·58-s + 84·64-s + 24·71-s − 160·72-s + 30·81-s − 120·92-s − 56·98-s − 200·100-s + 240·116-s + 16·121-s + 127-s − 120·128-s + 131-s + ⋯
L(s)  = 1  − 2.82·2-s + 5·4-s − 7.07·8-s + 8/3·9-s + 35/4·16-s − 7.54·18-s − 2.50·23-s − 4·25-s + 4.45·29-s − 9.89·32-s + 40/3·36-s + 7.07·46-s + 2·49-s + 11.3·50-s − 12.6·58-s + 21/2·64-s + 2.84·71-s − 18.8·72-s + 10/3·81-s − 12.5·92-s − 5.65·98-s − 20·100-s + 22.2·116-s + 1.45·121-s + 0.0887·127-s − 10.6·128-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(43.7050\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{322} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5712329982\)
\(L(\frac12)\) \(\approx\) \(0.5712329982\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 166 T^{2} + p^{2} T^{4} )^{2} \)
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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

−8.327295884818461448551047948773, −8.246808524132898051611036485790, −8.122299399584182675439812938478, −7.75599911046148835035416523604, −7.65877560421752824588729103866, −7.17715271022436983984392441292, −7.02029591069777022514908896740, −7.00860497571241306211756392789, −6.46639347526639826809600464490, −6.42358553718720914610377248831, −5.87308308284396725950810075422, −5.81447876427472224402836804844, −5.74522891009708330016810871975, −4.75127959649727462015182895875, −4.66772228430256113542159063911, −4.42682126234738428855271684586, −3.78545433701755671865717783205, −3.76856629766051743070373772046, −3.46409753294503503203626545132, −2.46226424312838764121391980359, −2.40418026451990752365784642526, −2.10381351862526570217413915343, −1.58868820700475365762621695545, −1.21919702188197437885948785301, −0.59765340142038398772023628798, 0.59765340142038398772023628798, 1.21919702188197437885948785301, 1.58868820700475365762621695545, 2.10381351862526570217413915343, 2.40418026451990752365784642526, 2.46226424312838764121391980359, 3.46409753294503503203626545132, 3.76856629766051743070373772046, 3.78545433701755671865717783205, 4.42682126234738428855271684586, 4.66772228430256113542159063911, 4.75127959649727462015182895875, 5.74522891009708330016810871975, 5.81447876427472224402836804844, 5.87308308284396725950810075422, 6.42358553718720914610377248831, 6.46639347526639826809600464490, 7.00860497571241306211756392789, 7.02029591069777022514908896740, 7.17715271022436983984392441292, 7.65877560421752824588729103866, 7.75599911046148835035416523604, 8.122299399584182675439812938478, 8.246808524132898051611036485790, 8.327295884818461448551047948773

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