Properties

Degree 8
Conductor $ 2^{20} \cdot 5^{12} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·9-s + 2·11-s − 2·19-s − 2·41-s + 49-s − 2·59-s + 10·81-s + 2·89-s − 8·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 8·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 4·9-s + 2·11-s − 2·19-s − 2·41-s + 49-s − 2·59-s + 10·81-s + 2·89-s − 8·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 8·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{20} \cdot 5^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{4000} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{20} \cdot 5^{12} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.1210823462\)
\(L(\frac12)\)  \(\approx\)  \(0.1210823462\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2$ \( ( 1 + T^{2} )^{4} \)
7$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
11$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
13$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
41$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
53$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
59$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
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

−6.23723442122574676473765063285, −5.83882136065378387837589732836, −5.75608960533784646845657061716, −5.65174714330096384006790224702, −5.57721357462453009047090832336, −5.15339221181815923094476856809, −5.08971323717827158839558591371, −4.62740817848954176014717518225, −4.62108847752277971524425784973, −4.48465535260203657476214622750, −4.16410299623851445200865800315, −3.79257508186621277055952131660, −3.66849357286705439055417409692, −3.39993831743869432911254259836, −3.27945918539258542339187992909, −3.25454378175731826934697389640, −2.76888523229911173567052149398, −2.63438398402300114061068444108, −2.27878941582784016870294192260, −2.16660993313762805158250279725, −1.97396195465770005998945690585, −1.63929935785592869824700844551, −1.04841541239011216516111618840, −1.00443465304627483002935169179, −0.13130051037305364865967451466, 0.13130051037305364865967451466, 1.00443465304627483002935169179, 1.04841541239011216516111618840, 1.63929935785592869824700844551, 1.97396195465770005998945690585, 2.16660993313762805158250279725, 2.27878941582784016870294192260, 2.63438398402300114061068444108, 2.76888523229911173567052149398, 3.25454378175731826934697389640, 3.27945918539258542339187992909, 3.39993831743869432911254259836, 3.66849357286705439055417409692, 3.79257508186621277055952131660, 4.16410299623851445200865800315, 4.48465535260203657476214622750, 4.62108847752277971524425784973, 4.62740817848954176014717518225, 5.08971323717827158839558591371, 5.15339221181815923094476856809, 5.57721357462453009047090832336, 5.65174714330096384006790224702, 5.75608960533784646845657061716, 5.83882136065378387837589732836, 6.23723442122574676473765063285

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