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Properties

Label 2-500-100.31-c0-0-0
Degree $2$
Conductor $500$
Sign $0.0627 + 0.998i$
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  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.5 − 1.53i)13-s + (0.309 − 0.951i)16-s + (0.5 + 0.363i)17-s − 0.999·18-s − 1.61·26-s + (−0.5 + 0.363i)29-s − 32-s + (0.190 − 0.587i)34-s + (0.309 + 0.951i)36-s + (−0.190 + 0.587i)37-s + (−0.5 + 1.53i)41-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.5 − 1.53i)13-s + (0.309 − 0.951i)16-s + (0.5 + 0.363i)17-s − 0.999·18-s − 1.61·26-s + (−0.5 + 0.363i)29-s − 32-s + (0.190 − 0.587i)34-s + (0.309 + 0.951i)36-s + (−0.190 + 0.587i)37-s + (−0.5 + 1.53i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.0627 + 0.998i$
Analytic conductor: \(0.249532\)
Root analytic conductor: \(0.499532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ 0.0627 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7273843490\)
\(L(\frac12)\) \(\approx\) \(0.7273843490\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.84774166270524732986561294994, −10.12074576301155721104951675345, −9.360124283208781505800553011247, −8.381273330474014809577558476669, −7.62933430497668731439454452980, −6.25660396517568648205399644964, −5.10011139322546652559662997938, −3.78052381227207248570949005381, −2.99970536277783307009686114347, −1.22639904476965981640667428583, 1.83690840557181911574673808019, 3.92322853445960267329599222447, 4.88940007036772082646059296898, 5.88680666185041904942053912371, 6.95358196106255122562595302646, 7.62044951110699001140253523361, 8.656321662076098581659311550598, 9.377083223750828429918196318035, 10.32267775904435384536196487069, 11.16197221951158132397163667987

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