Properties

Degree $8$
Conductor $203928109056$
Sign $1$
Motivic weight $1$
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 + 8·9-s + 8·11-s + 8·13-s − 16·23-s + 8·25-s + 12·27-s + 32·33-s + 8·37-s + 32·39-s − 16·47-s − 2·49-s − 8·59-s + 40·61-s − 64·69-s − 24·71-s − 24·73-s + 32·75-s + 23·81-s + 40·83-s − 24·97-s + 64·99-s + 24·107-s + 24·109-s + 32·111-s + 64·117-s − 4·121-s + ⋯
L(s)  = 1  + 2.30·3-s + 8/3·9-s + 2.41·11-s + 2.21·13-s − 3.33·23-s + 8/5·25-s + 2.30·27-s + 5.57·33-s + 1.31·37-s + 5.12·39-s − 2.33·47-s − 2/7·49-s − 1.04·59-s + 5.12·61-s − 7.70·69-s − 2.84·71-s − 2.80·73-s + 3.69·75-s + 23/9·81-s + 4.39·83-s − 2.43·97-s + 6.43·99-s + 2.32·107-s + 2.29·109-s + 3.03·111-s + 5.91·117-s − 0.363·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{672} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.5150\)
\(L(\frac12)\) \(\approx\) \(10.5150\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$D_4\times C_2$ \( 1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
13$D_{4}$ \( ( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 32 T^{2} + 690 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
29$D_4\times C_2$ \( 1 - 92 T^{2} + 3670 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
43$D_4\times C_2$ \( 1 - 20 T^{2} + 2646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 52 T^{2} + 4246 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 4 T + 104 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 4 T^{2} + 6934 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 - 20 T + 264 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 284 T^{2} + 35494 T^{4} - 284 p^{2} T^{6} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.85247088717976281955677758442, −7.23002652534497141722947204136, −7.15114864368907221882507410314, −7.02056175251060120140440101471, −6.51197269289063589164076919353, −6.35407543575744745154962984332, −6.26809706881516065164723057622, −6.11411524484603789493411544650, −5.78277286514227101071816200424, −5.53793252578151470096807336824, −4.96607079726573133850491214520, −4.62350095383187703664797220501, −4.62243800885960709008629911594, −4.07590662791715287407549147156, −3.86415380415872231759763070043, −3.82360681883261776225163036186, −3.52871098273680344426740647452, −3.39169834508120182371077619166, −2.93753438758147736786661562843, −2.63341935216332842441543841242, −2.28729350308410129217225848829, −1.78610389225483470104718565804, −1.69348668093072361580232521910, −1.25594799337853660265011980218, −0.810101194556645063547999786689, 0.810101194556645063547999786689, 1.25594799337853660265011980218, 1.69348668093072361580232521910, 1.78610389225483470104718565804, 2.28729350308410129217225848829, 2.63341935216332842441543841242, 2.93753438758147736786661562843, 3.39169834508120182371077619166, 3.52871098273680344426740647452, 3.82360681883261776225163036186, 3.86415380415872231759763070043, 4.07590662791715287407549147156, 4.62243800885960709008629911594, 4.62350095383187703664797220501, 4.96607079726573133850491214520, 5.53793252578151470096807336824, 5.78277286514227101071816200424, 6.11411524484603789493411544650, 6.26809706881516065164723057622, 6.35407543575744745154962984332, 6.51197269289063589164076919353, 7.02056175251060120140440101471, 7.15114864368907221882507410314, 7.23002652534497141722947204136, 7.85247088717976281955677758442

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