Properties

Degree 8
Conductor $ 2^{20} \cdot 7^{4} $
Sign $1$
Motivic weight 0
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 − 9-s − 2·17-s + 3·25-s + 2·37-s + 2·45-s − 2·49-s + 2·53-s + 2·61-s + 2·73-s + 81-s + 4·85-s + 2·89-s − 2·101-s − 2·109-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2·5-s − 9-s − 2·17-s + 3·25-s + 2·37-s + 2·45-s − 2·49-s + 2·53-s + 2·61-s + 2·73-s + 81-s + 4·85-s + 2·89-s − 2·101-s − 2·109-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{20} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{224} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{20} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.1812040511\)
\(L(\frac12)\)  \(\approx\)  \(0.1812040511\)
\(L(1)\)   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$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
19$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
59$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
79$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−9.309996863911135671874973673591, −8.897422967595805715172683466262, −8.499984918113276897785770765506, −8.294787861306813368008114044733, −8.215630286176052104965475202038, −8.152010468097277936804154069629, −7.64951924974791220569551434192, −7.30376574637874454210651227516, −7.10972948766242002299187214914, −6.96509250196731627601181733578, −6.37343180237093145708937355264, −6.27309012268999072264489363596, −6.25005304233019844508058994269, −5.57705880420581847333263358842, −5.04202855766530028312325826175, −4.97377363559341987547388628072, −4.86158159448459267204310288019, −4.23739228702478708277672850209, −3.85082022785688915825160501435, −3.84728006389649320724127329225, −3.51637984074493596937248463765, −2.76788953705767214556321314744, −2.50306402926853600546352933742, −2.37913706468566554218222193660, −1.18115060432250576025081445514, 1.18115060432250576025081445514, 2.37913706468566554218222193660, 2.50306402926853600546352933742, 2.76788953705767214556321314744, 3.51637984074493596937248463765, 3.84728006389649320724127329225, 3.85082022785688915825160501435, 4.23739228702478708277672850209, 4.86158159448459267204310288019, 4.97377363559341987547388628072, 5.04202855766530028312325826175, 5.57705880420581847333263358842, 6.25005304233019844508058994269, 6.27309012268999072264489363596, 6.37343180237093145708937355264, 6.96509250196731627601181733578, 7.10972948766242002299187214914, 7.30376574637874454210651227516, 7.64951924974791220569551434192, 8.152010468097277936804154069629, 8.215630286176052104965475202038, 8.294787861306813368008114044733, 8.499984918113276897785770765506, 8.897422967595805715172683466262, 9.309996863911135671874973673591

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