Properties

Degree 4
Conductor $ 2^{10} \cdot 5^{4} $
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  + 9-s + 2·11-s − 2·19-s − 2·41-s − 2·49-s + 4·59-s + 2·89-s + 2·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 2·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 9-s + 2·11-s − 2·19-s − 2·41-s − 2·49-s + 4·59-s + 2·89-s + 2·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 2·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(640000\)    =    \(2^{10} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{800} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 640000,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.048754437$
$L(\frac12)$  $\approx$  $1.048754437$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 - T^{2} + T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2^2$ \( 1 - T^{2} + T^{4} \)
19$C_2$ \( ( 1 + T + T^{2} )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_2$ \( ( 1 + T + T^{2} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$ \( ( 1 - T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2^2$ \( 1 - T^{2} + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 - T^{2} + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 - T^{2} + T^{4} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.48192462057179431669655267948, −10.26471179692433523487791927714, −9.888970010120026042719453466201, −9.433726876346548985088267965846, −8.974148495807358145771356534967, −8.524658810398930347786710734400, −8.360859442106513473343411294213, −7.66976433374811804530926445176, −7.05938144605083630950688159385, −6.76979535282880298690896417891, −6.30808991530566763121933921838, −6.24893824838174372412763924405, −5.13329225274981955757854630763, −4.96042355551600179416120835091, −4.14552487292280802101351485904, −3.89058780516858376452091421139, −3.54888068522024869069231136553, −2.49032747626348307530879777677, −1.84068117969644956727528700491, −1.27683073765345448383954374058, 1.27683073765345448383954374058, 1.84068117969644956727528700491, 2.49032747626348307530879777677, 3.54888068522024869069231136553, 3.89058780516858376452091421139, 4.14552487292280802101351485904, 4.96042355551600179416120835091, 5.13329225274981955757854630763, 6.24893824838174372412763924405, 6.30808991530566763121933921838, 6.76979535282880298690896417891, 7.05938144605083630950688159385, 7.66976433374811804530926445176, 8.360859442106513473343411294213, 8.524658810398930347786710734400, 8.974148495807358145771356534967, 9.433726876346548985088267965846, 9.888970010120026042719453466201, 10.26471179692433523487791927714, 10.48192462057179431669655267948

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