Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{2} $
Sign $0.447 + 0.894i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  i·3-s + 11-s i·17-s − 19-s i·27-s i·33-s − 41-s + 2i·43-s − 49-s − 51-s + i·57-s + 2·59-s + i·67-s + i·73-s − 81-s + ⋯
L(s)  = 1  i·3-s + 11-s i·17-s − 19-s i·27-s i·33-s − 41-s + 2i·43-s − 49-s − 51-s + i·57-s + 2·59-s + i·67-s + i·73-s − 81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(800\)    =    \(2^{5} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.447 + 0.894i$
motivic weight  =  \(0\)
character  :  $\chi_{800} (399, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 800,\ (\ :0),\ 0.447 + 0.894i)$
$L(\frac{1}{2})$  $\approx$  $1.024087124$
$L(\frac12)$  $\approx$  $1.024087124$
$L(1)$   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 2. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT - T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 - 2iT - T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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