Properties

Label 8-2e20-1.1-c1e4-0-0
Degree $8$
Conductor $1048576$
Sign $1$
Analytic cond. $0.00426293$
Root an. cond. $0.505491$
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  − 4·4-s − 4·5-s + 4·7-s − 2·9-s − 8·11-s + 4·13-s + 12·16-s − 8·19-s + 16·20-s + 12·23-s + 10·25-s + 8·27-s − 16·28-s − 4·29-s − 16·31-s − 16·35-s + 8·36-s + 4·37-s − 12·41-s + 16·43-s + 32·44-s + 8·45-s + 8·49-s − 16·52-s + 4·53-s + 32·55-s − 16·59-s + ⋯
L(s)  = 1  − 2·4-s − 1.78·5-s + 1.51·7-s − 2/3·9-s − 2.41·11-s + 1.10·13-s + 3·16-s − 1.83·19-s + 3.57·20-s + 2.50·23-s + 2·25-s + 1.53·27-s − 3.02·28-s − 0.742·29-s − 2.87·31-s − 2.70·35-s + 4/3·36-s + 0.657·37-s − 1.87·41-s + 2.43·43-s + 4.82·44-s + 1.19·45-s + 8/7·49-s − 2.21·52-s + 0.549·53-s + 4.31·55-s − 2.08·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(1048576\)    =    \(2^{20}\)
Sign: $1$
Analytic conductor: \(0.00426293\)
Root analytic conductor: \(0.505491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{32} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 1048576,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1758072183\)
\(L(\frac12)\) \(\approx\) \(0.1758072183\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \)
5$C_2$$\times$$C_2^2$ \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 8 T^{2} + p^{2} T^{4} ) \)
7$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \)
13$D_4\times C_2$ \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 8 T + 18 T^{2} - 160 T^{3} - 1246 T^{4} - 160 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 300 T^{3} + 1246 T^{4} - 300 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 4 T + 6 T^{2} - 204 T^{3} - 830 T^{4} - 204 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 3694 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 16 T + 162 T^{2} - 1384 T^{3} + 10178 T^{4} - 1384 p T^{5} + 162 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 4 T + 54 T^{2} - 708 T^{3} + 3490 T^{4} - 708 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 16 T + 114 T^{2} + 696 T^{3} + 4834 T^{4} + 696 p T^{5} + 114 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 8 T + 18 T^{2} - 736 T^{3} - 5854 T^{4} - 736 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 + 12 T + 72 T^{2} + 876 T^{3} + 10654 T^{4} + 876 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 16 T + 114 T^{2} - 792 T^{3} + 6370 T^{4} - 792 p T^{5} + 114 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1582 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−12.78272848163143092840612750486, −12.56606151155746504788626927166, −12.43048245779470944023421298680, −11.72501160152618188573262362939, −11.41255396244891181359750871590, −10.80960534141751908502174829319, −10.72033945482938012257679558430, −10.64958536266315894369863785049, −10.60767725755493944117655527694, −9.411335294844249942239170838702, −9.091594694511778732231568128747, −8.964605063623333690421383505347, −8.506567126214782974584811450981, −8.204311529247583565337826771467, −8.039681780751416313585515708752, −7.64153009694769787491189295563, −7.34456963808696468241255981512, −6.66895946208138919800828600992, −5.76679555433252604126019488759, −5.36197067505443972357916132126, −4.97583751447258677706480813545, −4.75694766846941895631079157983, −4.12636245126938214718428657231, −3.63022828323184476776207771085, −2.90836281020401295116138215067, 2.90836281020401295116138215067, 3.63022828323184476776207771085, 4.12636245126938214718428657231, 4.75694766846941895631079157983, 4.97583751447258677706480813545, 5.36197067505443972357916132126, 5.76679555433252604126019488759, 6.66895946208138919800828600992, 7.34456963808696468241255981512, 7.64153009694769787491189295563, 8.039681780751416313585515708752, 8.204311529247583565337826771467, 8.506567126214782974584811450981, 8.964605063623333690421383505347, 9.091594694511778732231568128747, 9.411335294844249942239170838702, 10.60767725755493944117655527694, 10.64958536266315894369863785049, 10.72033945482938012257679558430, 10.80960534141751908502174829319, 11.41255396244891181359750871590, 11.72501160152618188573262362939, 12.43048245779470944023421298680, 12.56606151155746504788626927166, 12.78272848163143092840612750486

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