Properties

Degree 8
Conductor $ 2^{12} \cdot 3^{8} \cdot 5^{8} $
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-s + 9-s + 2·11-s − 8·19-s + 36-s − 4·41-s + 2·44-s + 2·49-s − 2·59-s − 64-s − 8·76-s + 4·89-s + 2·99-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 8·171-s + 173-s + ⋯
L(s)  = 1  + 4-s + 9-s + 2·11-s − 8·19-s + 36-s − 4·41-s + 2·44-s + 2·49-s − 2·59-s − 64-s − 8·76-s + 4·89-s + 2·99-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 8·171-s + 173-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{12} \cdot 3^{8} \cdot 5^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{1800} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{12} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $0.9506311319$
$L(\frac12)$  $\approx$  $0.9506311319$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
19$C_1$ \( ( 1 + T )^{8} \)
23$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_2$ \( ( 1 + T + T^{2} )^{4} \)
43$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
47$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
89$C_2$ \( ( 1 - T + T^{2} )^{4} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
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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

−6.72438027726173487667877908285, −6.56875302196552206070886735752, −6.35932534241057453707300763664, −6.30625675643013660697211699987, −6.16901194325622101442255250526, −6.06203510735435564077399046923, −5.71416103453631749697386136369, −5.12881101680909443395623677804, −5.01416789179383666717981765888, −4.87113747219715815344168604260, −4.46176799045250495728152669080, −4.28594526994680840127388930445, −4.17447292961991938419969427431, −4.10187096955129550011389501728, −3.80289844165556543901472994640, −3.56601698822310785228126100365, −3.32332478407927401159022863107, −2.76594477286899714820133867788, −2.56859065300355188838635233570, −2.25859564692299210050947729146, −1.97058776441899405761209405845, −1.77935328095653857602908539241, −1.76793121082152067002823293072, −1.42860540125976050099974465698, −0.45395563423841967327361832375, 0.45395563423841967327361832375, 1.42860540125976050099974465698, 1.76793121082152067002823293072, 1.77935328095653857602908539241, 1.97058776441899405761209405845, 2.25859564692299210050947729146, 2.56859065300355188838635233570, 2.76594477286899714820133867788, 3.32332478407927401159022863107, 3.56601698822310785228126100365, 3.80289844165556543901472994640, 4.10187096955129550011389501728, 4.17447292961991938419969427431, 4.28594526994680840127388930445, 4.46176799045250495728152669080, 4.87113747219715815344168604260, 5.01416789179383666717981765888, 5.12881101680909443395623677804, 5.71416103453631749697386136369, 6.06203510735435564077399046923, 6.16901194325622101442255250526, 6.30625675643013660697211699987, 6.35932534241057453707300763664, 6.56875302196552206070886735752, 6.72438027726173487667877908285

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