↑

Properties

Label 2-1620-180.79-c0-0-5
Degree $2$
Conductor $1620$
Sign $0.173 - 0.984i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999i·8-s + (0.499 + 0.866i)10-s + (−1.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s − i·17-s + 0.999i·20-s + (0.499 + 0.866i)25-s − 1.73·26-s + (0.866 − 1.5i)29-s + (−0.866 + 0.499i)32-s + (0.5 − 0.866i)34-s − 1.73i·37-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999i·8-s + (0.499 + 0.866i)10-s + (−1.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s − i·17-s + 0.999i·20-s + (0.499 + 0.866i)25-s − 1.73·26-s + (0.866 − 1.5i)29-s + (−0.866 + 0.499i)32-s + (0.5 − 0.866i)34-s − 1.73i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.996545349\)
\(L(\frac12)\) \(\approx\) \(1.996545349\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 + (-0.866 - 0.5i)T \)
good7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - 1.73T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.672569713723557349007118403917, −9.068301695837407338645673682603, −7.85274251888205119445411399133, −7.12912821880700978230009479616, −6.57532934662455015950951018342, −5.64464347314300908626297058920, −4.92498080651624038295448019293, −4.05065653436278199447280501935, −2.72277560087324283984708393983, −2.19635934830532508241228443941, 1.33298622868522307760368747257, 2.43109394178940768097480918369, 3.29767632926102987660041848069, 4.65344081520144472969315772992, 5.09482030385721969592910086986, 5.97924060921549831247525692325, 6.71960862007963364035735181838, 7.74891021902168217495423669631, 8.796296669779050106366435545177, 9.652296718854423198192364428344

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