Properties

Label 8-1960e4-1.1-c1e4-0-3
Degree $8$
Conductor $1.476\times 10^{13}$
Sign $1$
Analytic cond. $59997.4$
Root an. cond. $3.95609$
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  − 2·3-s + 2·5-s + 5·9-s + 4·11-s − 8·13-s − 4·15-s − 4·17-s + 14·23-s + 25-s − 10·27-s − 20·29-s − 4·31-s − 8·33-s + 16·39-s − 12·41-s + 12·43-s + 10·45-s + 12·47-s + 8·51-s + 8·55-s − 8·59-s + 2·61-s − 16·65-s + 6·67-s − 28·69-s − 48·71-s − 4·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 5/3·9-s + 1.20·11-s − 2.21·13-s − 1.03·15-s − 0.970·17-s + 2.91·23-s + 1/5·25-s − 1.92·27-s − 3.71·29-s − 0.718·31-s − 1.39·33-s + 2.56·39-s − 1.87·41-s + 1.82·43-s + 1.49·45-s + 1.75·47-s + 1.12·51-s + 1.07·55-s − 1.04·59-s + 0.256·61-s − 1.98·65-s + 0.733·67-s − 3.37·69-s − 5.69·71-s − 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(59997.4\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.877736991\)
\(L(\frac12)\) \(\approx\) \(1.877736991\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
7 \( 1 \)
good3$D_4\times C_2$ \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 4 T - 2 T^{2} + 16 T^{3} + 27 T^{4} + 16 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
17$C_4\times C_2$ \( 1 + 4 T + 10 T^{2} - 112 T^{3} - 525 T^{4} - 112 p T^{5} + 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^3$ \( 1 - 6 T^{2} - 325 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 14 T + 103 T^{2} - 658 T^{3} + 3612 T^{4} - 658 p T^{5} + 103 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 4 T - 42 T^{2} - 16 T^{3} + 1907 T^{4} - 16 p T^{5} - 42 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^3$ \( 1 - 42 T^{2} + 395 T^{4} - 42 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 6 T - 3 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 12 T + 46 T^{2} - 48 T^{3} + 627 T^{4} - 48 p T^{5} + 46 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 74 T^{2} + 2667 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 2 T - 87 T^{2} + 62 T^{3} + 4316 T^{4} + 62 p T^{5} - 87 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 6 T - 9 T^{2} + 534 T^{3} - 4876 T^{4} + 534 p T^{5} - 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
73$D_4\times C_2$ \( 1 + 4 T - 102 T^{2} - 112 T^{3} + 7427 T^{4} - 112 p T^{5} - 102 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
83$D_{4}$ \( ( 1 - 18 T + 229 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 22 T + 217 T^{2} + 22 p T^{3} + 228 p T^{4} + 22 p^{2} T^{5} + 217 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
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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

−6.60358104084779261442531334988, −6.43248918749161162828341245657, −5.86871739422377481973774961366, −5.82144114676232867457707226665, −5.64655965797221910996436744012, −5.60101351024758720380747272725, −5.42256140193026794703822297835, −4.94961150681286790626784460892, −4.80683897202288295978781444565, −4.63130382762962622319403544108, −4.60161804479991093767367404680, −4.15025242153619191721745797788, −3.99850507132720828686530795474, −3.66005570370460033093663051106, −3.62953137543969708399175099860, −3.02685226922706435411782475494, −2.95792377415495318627261965315, −2.55735138254886223937107574474, −2.39045374121638507053395795534, −1.82965507482235433738511637111, −1.73179142722232328469696462737, −1.65333404675059905987554940209, −1.22475454935689919639932097588, −0.58286117478084295658072423903, −0.34941546957558404290195694030, 0.34941546957558404290195694030, 0.58286117478084295658072423903, 1.22475454935689919639932097588, 1.65333404675059905987554940209, 1.73179142722232328469696462737, 1.82965507482235433738511637111, 2.39045374121638507053395795534, 2.55735138254886223937107574474, 2.95792377415495318627261965315, 3.02685226922706435411782475494, 3.62953137543969708399175099860, 3.66005570370460033093663051106, 3.99850507132720828686530795474, 4.15025242153619191721745797788, 4.60161804479991093767367404680, 4.63130382762962622319403544108, 4.80683897202288295978781444565, 4.94961150681286790626784460892, 5.42256140193026794703822297835, 5.60101351024758720380747272725, 5.64655965797221910996436744012, 5.82144114676232867457707226665, 5.86871739422377481973774961366, 6.43248918749161162828341245657, 6.60358104084779261442531334988

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