↑

Properties

Label 2-500-100.59-c0-0-0
Degree $2$
Conductor $500$
Sign $0.837 - 0.545i$
Analytic cond. $0.249532$
Root an. cond. $0.499532$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (−0.363 − 0.5i)13-s + (−0.809 + 0.587i)16-s + (1.53 + 0.5i)17-s + i·18-s + 0.618·26-s + (0.5 + 1.53i)29-s − i·32-s + (−1.30 + 0.951i)34-s + (−0.809 − 0.587i)36-s + (−0.951 − 1.30i)37-s + (−0.5 + 0.363i)41-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (−0.363 − 0.5i)13-s + (−0.809 + 0.587i)16-s + (1.53 + 0.5i)17-s + i·18-s + 0.618·26-s + (0.5 + 1.53i)29-s − i·32-s + (−1.30 + 0.951i)34-s + (−0.809 − 0.587i)36-s + (−0.951 − 1.30i)37-s + (−0.5 + 0.363i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.837 - 0.545i$
Analytic conductor: \(0.249532\)
Root analytic conductor: \(0.499532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ 0.837 - 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6904099540\)
\(L(\frac12)\) \(\approx\) \(0.6904099540\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 \)
good3 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
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

−10.86215408427658291363431970357, −10.13036204883729870031236846872, −9.467119255212504422281469811435, −8.459209681568862562524150015795, −7.57221273158934059614937482438, −6.82302987146862284040218568718, −5.77871749933435305198277841174, −4.84827839873407239116580585402, −3.47032106328290932410879036961, −1.41330865428498230129814384147, 1.55826984972152016210647205189, 2.90684191476287605967796667377, 4.16244492629231719685796320247, 5.15357716706243678215344330818, 6.77954714133420550426039366108, 7.68558583920176380499027811157, 8.384748441772233415484447439056, 9.762664914547284891790109859098, 9.913805921746523793473888700613, 11.00594058134672001012692292101

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