Properties

Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·5-s + 9-s − 4·11-s − 16·13-s − 8·15-s + 4·17-s − 4·23-s + 12·25-s + 2·27-s − 16·29-s + 8·31-s + 8·33-s + 8·37-s + 32·39-s − 8·41-s + 4·45-s − 8·51-s + 4·53-s − 16·55-s + 8·59-s + 16·61-s − 64·65-s + 8·69-s + 8·71-s + 8·73-s − 24·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 1/3·9-s − 1.20·11-s − 4.43·13-s − 2.06·15-s + 0.970·17-s − 0.834·23-s + 12/5·25-s + 0.384·27-s − 2.97·29-s + 1.43·31-s + 1.39·33-s + 1.31·37-s + 5.12·39-s − 1.24·41-s + 0.596·45-s − 1.12·51-s + 0.549·53-s − 2.15·55-s + 1.04·59-s + 2.04·61-s − 7.93·65-s + 0.963·69-s + 0.949·71-s + 0.936·73-s − 2.77·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9888696366\)
\(L(\frac12)\) \(\approx\) \(0.9888696366\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_{4}$ \( ( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 4 T - 4 T^{2} + 56 T^{3} - 161 T^{4} + 56 p T^{5} - 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^3$ \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 4 T - 2 T^{2} - 112 T^{3} - 573 T^{4} - 112 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 8 T - 6 T^{2} - 64 T^{3} + 1955 T^{4} - 64 p T^{5} - 6 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 8 T - 62 T^{2} - 64 T^{3} + 8619 T^{4} - 64 p T^{5} - 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 16 T + 88 T^{2} - 736 T^{3} + 8887 T^{4} - 736 p T^{5} + 88 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^3$ \( 1 - 102 T^{2} + 5915 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 8 T + 656 T^{3} - 5905 T^{4} + 656 p T^{5} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 16 T + 66 T^{2} - 512 T^{3} + 9635 T^{4} - 512 p T^{5} + 66 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 20 T + 140 T^{2} + 1640 T^{3} + 24079 T^{4} + 1640 p T^{5} + 140 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 8 T + 208 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.29576108767144463025188781868, −6.04955579194452940198268365749, −5.84826869413173104508886197177, −5.59952470738727101180994847057, −5.59923126544920025429837741171, −5.30715942788641762204539344097, −5.25322973855090443721656628348, −4.95220607381280058497057720362, −4.94784064255828876783227819064, −4.66046665014611978795504974966, −4.31227277557701397878383861113, −4.30013901358592066771674641553, −3.69780783483002676894171138462, −3.66798169541418190122368433306, −3.15355701075482234813928683374, −2.98599309384768028519015982172, −2.69187251322358883402885896332, −2.34750664456240727557624162190, −2.33337366074546103979356501363, −2.16718439863959993082772461509, −1.92492230425792169912528662613, −1.47381991701470758275874543860, −0.993386080319218760693290486397, −0.53943860295613094382426913866, −0.25676550043747858011932985074, 0.25676550043747858011932985074, 0.53943860295613094382426913866, 0.993386080319218760693290486397, 1.47381991701470758275874543860, 1.92492230425792169912528662613, 2.16718439863959993082772461509, 2.33337366074546103979356501363, 2.34750664456240727557624162190, 2.69187251322358883402885896332, 2.98599309384768028519015982172, 3.15355701075482234813928683374, 3.66798169541418190122368433306, 3.69780783483002676894171138462, 4.30013901358592066771674641553, 4.31227277557701397878383861113, 4.66046665014611978795504974966, 4.94784064255828876783227819064, 4.95220607381280058497057720362, 5.25322973855090443721656628348, 5.30715942788641762204539344097, 5.59923126544920025429837741171, 5.59952470738727101180994847057, 5.84826869413173104508886197177, 6.04955579194452940198268365749, 6.29576108767144463025188781868

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