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Properties

Label 2-500-500.431-c0-0-0
Degree $2$
Conductor $500$
Sign $0.997 - 0.0753i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.929 + 0.368i)2-s + (0.728 − 0.684i)4-s + (0.876 − 0.481i)5-s + (−0.425 + 0.904i)8-s + (0.535 + 0.844i)9-s + (−0.637 + 0.770i)10-s + (−1.06 − 1.67i)13-s + (0.0627 − 0.998i)16-s + (0.781 + 0.733i)17-s + (−0.809 − 0.587i)18-s + (0.309 − 0.951i)20-s + (0.535 − 0.844i)25-s + (1.60 + 1.16i)26-s + (−0.0235 + 0.123i)29-s + (0.309 + 0.951i)32-s + ⋯
L(s)  = 1  + (−0.929 + 0.368i)2-s + (0.728 − 0.684i)4-s + (0.876 − 0.481i)5-s + (−0.425 + 0.904i)8-s + (0.535 + 0.844i)9-s + (−0.637 + 0.770i)10-s + (−1.06 − 1.67i)13-s + (0.0627 − 0.998i)16-s + (0.781 + 0.733i)17-s + (−0.809 − 0.587i)18-s + (0.309 − 0.951i)20-s + (0.535 − 0.844i)25-s + (1.60 + 1.16i)26-s + (−0.0235 + 0.123i)29-s + (0.309 + 0.951i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.997 - 0.0753i$
Analytic conductor: \(0.249532\)
Root analytic conductor: \(0.499532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ 0.997 - 0.0753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6872001477\)
\(L(\frac12)\) \(\approx\) \(0.6872001477\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.929 - 0.368i)T \)
5 \( 1 + (-0.876 + 0.481i)T \)
good3 \( 1 + (-0.535 - 0.844i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.728 + 0.684i)T^{2} \)
13 \( 1 + (1.06 + 1.67i)T + (-0.425 + 0.904i)T^{2} \)
17 \( 1 + (-0.781 - 0.733i)T + (0.0627 + 0.998i)T^{2} \)
19 \( 1 + (-0.535 + 0.844i)T^{2} \)
23 \( 1 + (0.187 - 0.982i)T^{2} \)
29 \( 1 + (0.0235 - 0.123i)T + (-0.929 - 0.368i)T^{2} \)
31 \( 1 + (-0.0627 - 0.998i)T^{2} \)
37 \( 1 + (0.0800 - 1.27i)T + (-0.992 - 0.125i)T^{2} \)
41 \( 1 + (0.929 - 1.12i)T + (-0.187 - 0.982i)T^{2} \)
43 \( 1 + (0.809 + 0.587i)T^{2} \)
47 \( 1 + (0.637 - 0.770i)T^{2} \)
53 \( 1 + (-0.371 + 0.0469i)T + (0.968 - 0.248i)T^{2} \)
59 \( 1 + (-0.876 + 0.481i)T^{2} \)
61 \( 1 + (1.11 + 1.35i)T + (-0.187 + 0.982i)T^{2} \)
67 \( 1 + (0.929 - 0.368i)T^{2} \)
71 \( 1 + (0.637 - 0.770i)T^{2} \)
73 \( 1 + (0.824 + 0.211i)T + (0.876 + 0.481i)T^{2} \)
79 \( 1 + (-0.535 - 0.844i)T^{2} \)
83 \( 1 + (-0.535 + 0.844i)T^{2} \)
89 \( 1 + (1.80 + 0.462i)T + (0.876 + 0.481i)T^{2} \)
97 \( 1 + (0.362 - 1.90i)T + (-0.929 - 0.368i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.60825054416984185175716598355, −10.18194605132730330684858545486, −9.561521778680748371786555445171, −8.280550430932658080810216650803, −7.84137232159946957792765311517, −6.68717900629209500753783292728, −5.55605963900150652958108742160, −4.97019943729528301590689326152, −2.80092787215675148279023837085, −1.50190675476821892391134784656, 1.68248539810896953660076410039, 2.83574462172035943458752734831, 4.19065122839797034778354082879, 5.79627205837351747009582492874, 7.00235164432109282819652578215, 7.22190409501019629680517631365, 8.855262656026323581621122492934, 9.517287429501834941972027568001, 9.972386203917803751810943423152, 10.98004863701524718749734189832

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