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Properties

Label 2-500-500.131-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  + (0.968 + 0.248i)2-s + (0.876 + 0.481i)4-s + (−0.187 − 0.982i)5-s + (0.728 + 0.684i)8-s + (−0.929 − 0.368i)9-s + (0.0627 − 0.998i)10-s + (0.791 + 0.313i)13-s + (0.535 + 0.844i)16-s + (−1.62 + 0.895i)17-s + (−0.809 − 0.587i)18-s + (0.309 − 0.951i)20-s + (−0.929 + 0.368i)25-s + (0.688 + 0.500i)26-s + (−1.06 + 0.134i)29-s + (0.309 + 0.951i)32-s + ⋯
L(s)  = 1  + (0.968 + 0.248i)2-s + (0.876 + 0.481i)4-s + (−0.187 − 0.982i)5-s + (0.728 + 0.684i)8-s + (−0.929 − 0.368i)9-s + (0.0627 − 0.998i)10-s + (0.791 + 0.313i)13-s + (0.535 + 0.844i)16-s + (−1.62 + 0.895i)17-s + (−0.809 − 0.587i)18-s + (0.309 − 0.951i)20-s + (−0.929 + 0.368i)25-s + (0.688 + 0.500i)26-s + (−1.06 + 0.134i)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: $\chi_{500} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ 0.997 - 0.0753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.411328279\)
\(L(\frac12)\) \(\approx\) \(1.411328279\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.968 - 0.248i)T \)
5 \( 1 + (0.187 + 0.982i)T \)
good3 \( 1 + (0.929 + 0.368i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.876 - 0.481i)T^{2} \)
13 \( 1 + (-0.791 - 0.313i)T + (0.728 + 0.684i)T^{2} \)
17 \( 1 + (1.62 - 0.895i)T + (0.535 - 0.844i)T^{2} \)
19 \( 1 + (0.929 - 0.368i)T^{2} \)
23 \( 1 + (0.992 - 0.125i)T^{2} \)
29 \( 1 + (1.06 - 0.134i)T + (0.968 - 0.248i)T^{2} \)
31 \( 1 + (-0.535 + 0.844i)T^{2} \)
37 \( 1 + (-0.0672 - 0.106i)T + (-0.425 + 0.904i)T^{2} \)
41 \( 1 + (-0.110 + 1.74i)T + (-0.992 - 0.125i)T^{2} \)
43 \( 1 + (0.809 + 0.587i)T^{2} \)
47 \( 1 + (-0.0627 + 0.998i)T^{2} \)
53 \( 1 + (-0.844 - 1.79i)T + (-0.637 + 0.770i)T^{2} \)
59 \( 1 + (0.187 + 0.982i)T^{2} \)
61 \( 1 + (0.0235 + 0.374i)T + (-0.992 + 0.125i)T^{2} \)
67 \( 1 + (-0.968 - 0.248i)T^{2} \)
71 \( 1 + (-0.0627 + 0.998i)T^{2} \)
73 \( 1 + (0.929 + 1.12i)T + (-0.187 + 0.982i)T^{2} \)
79 \( 1 + (0.929 + 0.368i)T^{2} \)
83 \( 1 + (0.929 - 0.368i)T^{2} \)
89 \( 1 + (1.23 + 1.49i)T + (-0.187 + 0.982i)T^{2} \)
97 \( 1 + (-1.26 + 0.159i)T + (0.968 - 0.248i)T^{2} \)
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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

−11.39452693360732258350485529137, −10.63738607372201708670509308317, −8.921191491745275521405796933063, −8.634399182181972544899028658038, −7.41395372999770276826961132803, −6.25076074727498978817655740474, −5.58090647155704131659080411695, −4.41237596802890582271516749450, −3.61892956205586714665238413647, −2.01781518817446259641593399392, 2.30116097857052367807428886908, 3.18491107612532695499241794037, 4.32346558075033571093308007983, 5.54180948192422727514426237565, 6.39337365058050508347952504677, 7.22625731666221305113619601754, 8.318067874780339737651529230068, 9.572748863629986057395281435582, 10.71835016981634340455075304756, 11.24975613726291990771498359231

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