↑

Properties

Label 2-500-500.371-c0-0-0
Degree $2$
Conductor $500$
Sign $-0.850 + 0.525i$
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  + (0.0627 − 0.998i)2-s + (−0.992 − 0.125i)4-s + (−0.425 − 0.904i)5-s + (−0.187 + 0.982i)8-s + (−0.637 − 0.770i)9-s + (−0.929 + 0.368i)10-s + (−1.11 − 1.35i)13-s + (0.968 + 0.248i)16-s + (1.26 − 0.159i)17-s + (−0.809 + 0.587i)18-s + (0.309 + 0.951i)20-s + (−0.637 + 0.770i)25-s + (−1.41 + 1.03i)26-s + (1.41 + 1.32i)29-s + (0.309 − 0.951i)32-s + ⋯
L(s)  = 1  + (0.0627 − 0.998i)2-s + (−0.992 − 0.125i)4-s + (−0.425 − 0.904i)5-s + (−0.187 + 0.982i)8-s + (−0.637 − 0.770i)9-s + (−0.929 + 0.368i)10-s + (−1.11 − 1.35i)13-s + (0.968 + 0.248i)16-s + (1.26 − 0.159i)17-s + (−0.809 + 0.587i)18-s + (0.309 + 0.951i)20-s + (−0.637 + 0.770i)25-s + (−1.41 + 1.03i)26-s + (1.41 + 1.32i)29-s + (0.309 − 0.951i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(0.249532\)
Root analytic conductor: \(0.499532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6719885242\)
\(L(\frac12)\) \(\approx\) \(0.6719885242\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0627 + 0.998i)T \)
5 \( 1 + (0.425 + 0.904i)T \)
good3 \( 1 + (0.637 + 0.770i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
11 \( 1 + (0.992 + 0.125i)T^{2} \)
13 \( 1 + (1.11 + 1.35i)T + (-0.187 + 0.982i)T^{2} \)
17 \( 1 + (-1.26 + 0.159i)T + (0.968 - 0.248i)T^{2} \)
19 \( 1 + (0.637 - 0.770i)T^{2} \)
23 \( 1 + (-0.728 - 0.684i)T^{2} \)
29 \( 1 + (-1.41 - 1.32i)T + (0.0627 + 0.998i)T^{2} \)
31 \( 1 + (-0.968 + 0.248i)T^{2} \)
37 \( 1 + (1.80 + 0.462i)T + (0.876 + 0.481i)T^{2} \)
41 \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (0.929 - 0.368i)T^{2} \)
53 \( 1 + (-1.27 + 0.702i)T + (0.535 - 0.844i)T^{2} \)
59 \( 1 + (0.425 + 0.904i)T^{2} \)
61 \( 1 + (-0.791 - 0.313i)T + (0.728 + 0.684i)T^{2} \)
67 \( 1 + (-0.0627 + 0.998i)T^{2} \)
71 \( 1 + (0.929 - 0.368i)T^{2} \)
73 \( 1 + (0.200 + 0.316i)T + (-0.425 + 0.904i)T^{2} \)
79 \( 1 + (0.637 + 0.770i)T^{2} \)
83 \( 1 + (0.637 - 0.770i)T^{2} \)
89 \( 1 + (-0.0672 - 0.106i)T + (-0.425 + 0.904i)T^{2} \)
97 \( 1 + (-0.781 - 0.733i)T + (0.0627 + 0.998i)T^{2} \)
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

−10.75543608097253754145468808136, −9.991048339995072758173393116320, −9.082380694980573665859300216477, −8.374035049585496865991781973249, −7.42510776539673387875060228709, −5.61606991733413439209691131234, −5.06612083502676656366016280307, −3.74256220772029141364095212280, −2.80558631459364303782571880132, −0.875338265131796993575081512371, 2.62524001360852216943637966430, 3.99379389593471757076926855596, 5.02796824843047625602209501575, 6.11244384900865818939839333919, 7.04579715038936953666923266319, 7.73779388606606243027798288351, 8.556885746363924325958561630000, 9.737962967334468883475682353065, 10.40642912315126819406622901811, 11.67945601946079515148224046366

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