Properties

Label 2-500-500.191-c0-0-0
Degree $2$
Conductor $500$
Sign $0.402 + 0.915i$
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  + (0.876 − 0.481i)2-s + (0.535 − 0.844i)4-s + (−0.929 − 0.368i)5-s + (0.0627 − 0.998i)8-s + (0.728 − 0.684i)9-s + (−0.992 + 0.125i)10-s + (−0.929 + 0.872i)13-s + (−0.425 − 0.904i)16-s + (0.781 + 1.23i)17-s + (0.309 − 0.951i)18-s + (−0.809 + 0.587i)20-s + (0.728 + 0.684i)25-s + (−0.393 + 1.21i)26-s + (−0.824 − 0.211i)29-s + (−0.809 − 0.587i)32-s + ⋯
L(s)  = 1  + (0.876 − 0.481i)2-s + (0.535 − 0.844i)4-s + (−0.929 − 0.368i)5-s + (0.0627 − 0.998i)8-s + (0.728 − 0.684i)9-s + (−0.992 + 0.125i)10-s + (−0.929 + 0.872i)13-s + (−0.425 − 0.904i)16-s + (0.781 + 1.23i)17-s + (0.309 − 0.951i)18-s + (−0.809 + 0.587i)20-s + (0.728 + 0.684i)25-s + (−0.393 + 1.21i)26-s + (−0.824 − 0.211i)29-s + (−0.809 − 0.587i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(0.249532\)
Root analytic conductor: \(0.499532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ 0.402 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.256453492\)
\(L(\frac12)\) \(\approx\) \(1.256453492\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.876 + 0.481i)T \)
5 \( 1 + (0.929 + 0.368i)T \)
good3 \( 1 + (-0.728 + 0.684i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
11 \( 1 + (-0.535 + 0.844i)T^{2} \)
13 \( 1 + (0.929 - 0.872i)T + (0.0627 - 0.998i)T^{2} \)
17 \( 1 + (-0.781 - 1.23i)T + (-0.425 + 0.904i)T^{2} \)
19 \( 1 + (-0.728 - 0.684i)T^{2} \)
23 \( 1 + (-0.968 - 0.248i)T^{2} \)
29 \( 1 + (0.824 + 0.211i)T + (0.876 + 0.481i)T^{2} \)
31 \( 1 + (0.425 - 0.904i)T^{2} \)
37 \( 1 + (-0.844 - 1.79i)T + (-0.637 + 0.770i)T^{2} \)
41 \( 1 + (1.06 - 0.134i)T + (0.968 - 0.248i)T^{2} \)
43 \( 1 + (-0.309 + 0.951i)T^{2} \)
47 \( 1 + (0.992 - 0.125i)T^{2} \)
53 \( 1 + (1.23 + 1.49i)T + (-0.187 + 0.982i)T^{2} \)
59 \( 1 + (0.929 + 0.368i)T^{2} \)
61 \( 1 + (-1.84 - 0.233i)T + (0.968 + 0.248i)T^{2} \)
67 \( 1 + (-0.876 + 0.481i)T^{2} \)
71 \( 1 + (0.992 - 0.125i)T^{2} \)
73 \( 1 + (0.0235 + 0.123i)T + (-0.929 + 0.368i)T^{2} \)
79 \( 1 + (-0.728 + 0.684i)T^{2} \)
83 \( 1 + (-0.728 - 0.684i)T^{2} \)
89 \( 1 + (0.328 + 1.72i)T + (-0.929 + 0.368i)T^{2} \)
97 \( 1 + (0.362 + 0.0931i)T + (0.876 + 0.481i)T^{2} \)
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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

−11.36290912119757999873271096769, −10.12184533574380495199770532954, −9.531035312622849805943993665545, −8.207795951932608861139227572163, −7.15720691335443332427059523582, −6.32889833170827018544446280104, −4.99092810148779155050946933560, −4.17317439495349933110254023257, −3.32887906439124947990127774348, −1.57786285725741538891877219249, 2.55255749585969842413646321168, 3.63026029367117938613158663860, 4.73262194946583383751823982928, 5.49215421847550422737571587987, 6.97097078066861598330095154218, 7.50963865539263680479678297604, 8.119436186935653428874083001686, 9.598372557249459331307448470336, 10.68463664629068779456110545823, 11.44467010617493900612808866396

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