Properties

Degree 8
Conductor $ 2^{8} \cdot 5^{4} $
Sign $1$
Motivic weight 2
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 + 16·9-s − 16·13-s − 24·17-s − 32·18-s + 10·25-s + 32·26-s − 8·29-s + 32·32-s + 48·34-s + 16·37-s − 112·41-s + 96·49-s − 20·50-s − 176·53-s + 16·58-s + 128·61-s − 64·64-s + 264·73-s − 32·74-s + 50·81-s + 224·82-s − 88·89-s − 264·97-s − 192·98-s + 328·101-s + 352·106-s + ⋯
L(s)  = 1  − 2-s + 16/9·9-s − 1.23·13-s − 1.41·17-s − 1.77·18-s + 2/5·25-s + 1.23·26-s − 0.275·29-s + 32-s + 1.41·34-s + 0.432·37-s − 2.73·41-s + 1.95·49-s − 2/5·50-s − 3.32·53-s + 8/29·58-s + 2.09·61-s − 64-s + 3.61·73-s − 0.432·74-s + 0.617·81-s + 2.73·82-s − 0.988·89-s − 2.72·97-s − 1.95·98-s + 3.24·101-s + 3.32·106-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(160000\)    =    \(2^{8} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{20} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 160000,\ (\ :1, 1, 1, 1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.342682$
$L(\frac12)$  $\approx$  $0.342682$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2$C_4$ \( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
good3$C_2^2:C_4$ \( 1 - 16 T^{2} + 206 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8} \)
7$C_2^2:C_4$ \( 1 - 96 T^{2} + 6606 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2^2:C_4$ \( 1 - 84 T^{2} - 7674 T^{4} - 84 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 + 8 T + 334 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 12 T + 294 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$C_2^2:C_4$ \( 1 - 1124 T^{2} + 571366 T^{4} - 1124 p^{4} T^{6} + p^{8} T^{8} \)
23$C_2^2:C_4$ \( 1 - 1856 T^{2} + 1404046 T^{4} - 1856 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 + 4 T + 1606 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$C_2^2:C_4$ \( 1 - 1524 T^{2} + 1236966 T^{4} - 1524 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 - 8 T + 2254 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 56 T + 3966 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$C_2^2:C_4$ \( 1 - 6896 T^{2} + 18665806 T^{4} - 6896 p^{4} T^{6} + p^{8} T^{8} \)
47$C_2^2:C_4$ \( 1 - 4736 T^{2} + 14725966 T^{4} - 4736 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 + 88 T + 7054 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$C_2^2:C_4$ \( 1 - 8164 T^{2} + 34261926 T^{4} - 8164 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 - 64 T + 5086 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2^2:C_4$ \( 1 - 7536 T^{2} + 47276046 T^{4} - 7536 p^{4} T^{6} + p^{8} T^{8} \)
71$C_2^2:C_4$ \( 1 - 12084 T^{2} + 81146406 T^{4} - 12084 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 132 T + 9894 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2^2:C_4$ \( 1 - 11844 T^{2} + 72414726 T^{4} - 11844 p^{4} T^{6} + p^{8} T^{8} \)
83$C_2^2:C_4$ \( 1 - 21296 T^{2} + 201120526 T^{4} - 21296 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 + 44 T + 8326 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 132 T + 22454 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.66048308923934595117485624503, −13.46485836927477789336601076842, −13.08663694101274076841807556316, −12.49875361328906704754609653084, −12.48887786565000405871751767505, −12.21303170706440393411827475958, −11.35638204847778194796798066602, −11.29415588817075777185017400195, −10.88861831861723776689182975920, −10.13985424320236302314452955734, −10.01272837542119754235070824034, −9.788602370359588837916928515174, −9.399943457714330104094633546675, −8.865872522679871094283943224714, −8.592821475085787209603572063168, −8.062354773420145265675760261931, −7.64047041750971698148875835846, −6.98214116978119855428849482800, −6.85329683828389638656960119829, −6.39638741873142387487651468247, −5.43441430703993799926017740945, −4.68474825363585902296547097666, −4.53806032610858065508113104593, −3.53233895242241329213728963533, −2.16603142707187025273094328595, 2.16603142707187025273094328595, 3.53233895242241329213728963533, 4.53806032610858065508113104593, 4.68474825363585902296547097666, 5.43441430703993799926017740945, 6.39638741873142387487651468247, 6.85329683828389638656960119829, 6.98214116978119855428849482800, 7.64047041750971698148875835846, 8.062354773420145265675760261931, 8.592821475085787209603572063168, 8.865872522679871094283943224714, 9.399943457714330104094633546675, 9.788602370359588837916928515174, 10.01272837542119754235070824034, 10.13985424320236302314452955734, 10.88861831861723776689182975920, 11.29415588817075777185017400195, 11.35638204847778194796798066602, 12.21303170706440393411827475958, 12.48887786565000405871751767505, 12.49875361328906704754609653084, 13.08663694101274076841807556316, 13.46485836927477789336601076842, 13.66048308923934595117485624503

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