Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 5^{2} $
Sign $0.917 - 0.397i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + 0.999·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + (0.866 + 0.499i)12-s + (−0.5 + 0.866i)16-s + i·17-s + (0.866 − 0.499i)18-s − 2·19-s + (0.866 − 0.499i)22-s + (0.499 + 0.866i)24-s − 0.999i·27-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + 0.999·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + (0.866 + 0.499i)12-s + (−0.5 + 0.866i)16-s + i·17-s + (0.866 − 0.499i)18-s − 2·19-s + (0.866 − 0.499i)22-s + (0.499 + 0.866i)24-s − 0.999i·27-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.917 - 0.397i$
motivic weight  =  \(0\)
character  :  $\chi_{1800} (1699, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1800,\ (\ :0),\ 0.917 - 0.397i)$
$L(\frac{1}{2})$  $\approx$  $2.447604839$
$L(\frac12)$  $\approx$  $2.447604839$
$L(1)$   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 2. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
good7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + 2T + T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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

−9.055366529591435716991555914793, −8.554004507572436256307676986787, −7.976497139364867146572044467996, −6.99913476647046270547739846305, −6.35932534241057453707300763664, −5.71416103453631749697386136369, −4.28594526994680840127388930445, −3.80289844165556543901472994640, −2.76594477286899714820133867788, −1.77935328095653857602908539241, 1.76793121082152067002823293072, 2.56859065300355188838635233570, 3.56601698822310785228126100365, 4.46176799045250495728152669080, 4.87113747219715815344168604260, 6.16901194325622101442255250526, 6.94794582753846297556298730463, 7.79990175788995072187836368129, 8.916934684660653139819644055051, 9.437697900466527160636995367458

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