Properties

Degree 8
Conductor $ 2^{20} \cdot 7^{4} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 17·9-s + 96·13-s − 2·17-s + 51·25-s + 96·29-s + 98·37-s − 192·41-s + 34·45-s − 98·49-s + 50·53-s + 2·61-s − 192·65-s − 190·73-s + 81·81-s + 4·85-s − 190·89-s + 576·97-s − 146·101-s + 142·109-s + 384·113-s − 1.63e3·117-s + 47·121-s − 198·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 2/5·5-s − 1.88·9-s + 7.38·13-s − 0.117·17-s + 2.03·25-s + 3.31·29-s + 2.64·37-s − 4.68·41-s + 0.755·45-s − 2·49-s + 0.943·53-s + 2/61·61-s − 2.95·65-s − 2.60·73-s + 81-s + 4/85·85-s − 2.13·89-s + 5.93·97-s − 1.44·101-s + 1.30·109-s + 3.39·113-s − 13.9·117-s + 0.388·121-s − 1.58·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{20} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{224} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{20} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(4.65814\)
\(L(\frac12)\)  \(\approx\)  \(4.65814\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
good3$C_2^3$ \( 1 + 17 T^{2} + 208 T^{4} + 17 p^{4} T^{6} + p^{8} T^{8} \)
5$C_2^2$ \( ( 1 + T - 24 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$C_2^3$ \( 1 - 47 T^{2} - 12432 T^{4} - 47 p^{4} T^{6} + p^{8} T^{8} \)
13$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + T - 288 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$C_2^3$ \( 1 + 673 T^{2} + 322608 T^{4} + 673 p^{4} T^{6} + p^{8} T^{8} \)
23$C_2^3$ \( 1 + 1009 T^{2} + 738240 T^{4} + 1009 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{4} \)
31$C_2^3$ \( 1 + 241 T^{2} - 865440 T^{4} + 241 p^{4} T^{6} + p^{8} T^{8} \)
37$C_2^2$ \( ( 1 - 49 T + 1032 T^{2} - 49 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 48 T + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 3122 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 1393 T^{2} - 2939232 T^{4} + 1393 p^{4} T^{6} + p^{8} T^{8} \)
53$C_2^2$ \( ( 1 - 25 T - 2184 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 6673 T^{2} + 32411568 T^{4} + 6673 p^{4} T^{6} + p^{8} T^{8} \)
61$C_2^2$ \( ( 1 - T - 3720 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 4753 T^{2} + 2439888 T^{4} + 4753 p^{4} T^{6} + p^{8} T^{8} \)
71$C_2^2$ \( ( 1 - 866 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 95 T + 3696 T^{2} + 95 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2^3$ \( 1 + 10801 T^{2} + 77711520 T^{4} + 10801 p^{4} T^{6} + p^{8} T^{8} \)
83$C_2^2$ \( ( 1 - 8594 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 95 T + 1104 T^{2} + 95 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 144 T + p^{2} T^{2} )^{4} \)
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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

−8.631523588927382222490841021281, −8.480687416624560540349841719584, −8.325856362904633612291798460258, −8.124221418843820805021395188875, −8.106572186596981669582269284407, −7.39233571834020532379012775333, −6.76453389539697833731534766449, −6.67258458886587069052492617636, −6.52037659626254115318793603819, −6.21939568193788991521427275247, −5.93200049368150219763611210724, −5.92970404734542269380602983807, −5.61482349692985381011255685271, −5.04927219900615403816340806687, −4.62780547793922233015425645479, −4.39927744703282405432001409888, −4.07116894065548118611616255955, −3.36540246700315755760902864489, −3.26480426475421587837184058987, −3.26141490713238760708481507861, −3.08224049187089788702005611500, −2.03452409131156817553221692859, −1.33638206571630237313296692738, −1.12026154097295026105710592308, −0.77767828322521165616781488047, 0.77767828322521165616781488047, 1.12026154097295026105710592308, 1.33638206571630237313296692738, 2.03452409131156817553221692859, 3.08224049187089788702005611500, 3.26141490713238760708481507861, 3.26480426475421587837184058987, 3.36540246700315755760902864489, 4.07116894065548118611616255955, 4.39927744703282405432001409888, 4.62780547793922233015425645479, 5.04927219900615403816340806687, 5.61482349692985381011255685271, 5.92970404734542269380602983807, 5.93200049368150219763611210724, 6.21939568193788991521427275247, 6.52037659626254115318793603819, 6.67258458886587069052492617636, 6.76453389539697833731534766449, 7.39233571834020532379012775333, 8.106572186596981669582269284407, 8.124221418843820805021395188875, 8.325856362904633612291798460258, 8.480687416624560540349841719584, 8.631523588927382222490841021281

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