Properties

Label 8-570e4-1.1-c1e4-0-1
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $429.148$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 4-s − 2·5-s − 4·6-s − 4·7-s − 2·8-s + 9-s − 4·10-s − 2·11-s − 2·12-s + 13-s − 8·14-s + 4·15-s − 4·16-s + 2·18-s + 3·19-s − 2·20-s + 8·21-s − 4·22-s − 23-s + 4·24-s + 25-s + 2·26-s + 2·27-s − 4·28-s + 12·29-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s − 1.51·7-s − 0.707·8-s + 1/3·9-s − 1.26·10-s − 0.603·11-s − 0.577·12-s + 0.277·13-s − 2.13·14-s + 1.03·15-s − 16-s + 0.471·18-s + 0.688·19-s − 0.447·20-s + 1.74·21-s − 0.852·22-s − 0.208·23-s + 0.816·24-s + 1/5·25-s + 0.392·26-s + 0.384·27-s − 0.755·28-s + 2.22·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(429.148\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1697636991\)
\(L(\frac12)\) \(\approx\) \(0.1697636991\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
3$C_2$ \( ( 1 + T + T^{2} )^{2} \)
5$C_2$ \( ( 1 + T + T^{2} )^{2} \)
19$C_2^2$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
good7$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
11$D_{4}$ \( ( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - T - 7 T^{2} + 18 T^{3} - 118 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + T - 27 T^{2} - 18 T^{3} + 232 T^{4} - 18 p T^{5} - 27 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
41$D_4\times C_2$ \( 1 + T - 63 T^{2} - 18 T^{3} + 2374 T^{4} - 18 p T^{5} - 63 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 9 T - 7 T^{2} + 18 T^{3} + 2412 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 11 T + 3 T^{2} + 132 T^{3} + 4702 T^{4} + 132 p T^{5} + 3 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 10 T + 30 T^{2} + 480 T^{3} - 4481 T^{4} + 480 p T^{5} + 30 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 3 T - 97 T^{2} + 48 T^{3} + 6966 T^{4} + 48 p T^{5} - 97 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 13 T + 11 T^{2} - 312 T^{3} + 9152 T^{4} - 312 p T^{5} + 11 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 + 2 T - 66 T^{2} - 144 T^{3} - 425 T^{4} - 144 p T^{5} - 66 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 15 T + 41 T^{2} - 570 T^{3} + 12102 T^{4} - 570 p T^{5} + 41 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 5 T + 25 T^{2} + 10 p T^{3} - 104 p T^{4} + 10 p^{2} T^{5} + 25 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
89$D_4\times C_2$ \( 1 + 9 T + 47 T^{2} - 1296 T^{3} - 13974 T^{4} - 1296 p T^{5} + 47 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.63692490956924695363476006758, −7.63486540773088237729388052755, −6.90386483072044435145287749766, −6.86685834437280169404905797608, −6.64390014301255089113725834474, −6.50482067623827917410427134437, −6.25294014105870489654570349695, −6.21572377067093524337183460088, −5.68611338796817555120309423138, −5.37490950381912048044220657753, −5.26059112806970958328214933517, −5.02278075545527508646029796417, −4.87892045185958998425650059452, −4.67675488311901983742177546034, −4.15240873729066086379028745080, −3.92043005706454492913084453699, −3.67021370753673792079380701801, −3.32721916202657845987984950242, −3.27889352658067559263446694536, −2.95744198429218754653464790257, −2.53392617694469497759118158751, −2.14941285181559990841856776590, −1.49379574705798438540401070417, −0.897036902763835848930715897183, −0.13528159176385247344993831928, 0.13528159176385247344993831928, 0.897036902763835848930715897183, 1.49379574705798438540401070417, 2.14941285181559990841856776590, 2.53392617694469497759118158751, 2.95744198429218754653464790257, 3.27889352658067559263446694536, 3.32721916202657845987984950242, 3.67021370753673792079380701801, 3.92043005706454492913084453699, 4.15240873729066086379028745080, 4.67675488311901983742177546034, 4.87892045185958998425650059452, 5.02278075545527508646029796417, 5.26059112806970958328214933517, 5.37490950381912048044220657753, 5.68611338796817555120309423138, 6.21572377067093524337183460088, 6.25294014105870489654570349695, 6.50482067623827917410427134437, 6.64390014301255089113725834474, 6.86685834437280169404905797608, 6.90386483072044435145287749766, 7.63486540773088237729388052755, 7.63692490956924695363476006758

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