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Properties

Label 2-500-500.291-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  + (−0.187 − 0.982i)2-s + (−0.929 + 0.368i)4-s + (0.968 − 0.248i)5-s + (0.535 + 0.844i)8-s + (0.876 + 0.481i)9-s + (−0.425 − 0.904i)10-s + (0.110 + 0.0604i)13-s + (0.728 − 0.684i)16-s + (−1.62 − 0.645i)17-s + (0.309 − 0.951i)18-s + (−0.809 + 0.587i)20-s + (0.876 − 0.481i)25-s + (0.0388 − 0.119i)26-s + (−0.929 − 1.12i)29-s + (−0.809 − 0.587i)32-s + ⋯
L(s)  = 1  + (−0.187 − 0.982i)2-s + (−0.929 + 0.368i)4-s + (0.968 − 0.248i)5-s + (0.535 + 0.844i)8-s + (0.876 + 0.481i)9-s + (−0.425 − 0.904i)10-s + (0.110 + 0.0604i)13-s + (0.728 − 0.684i)16-s + (−1.62 − 0.645i)17-s + (0.309 − 0.951i)18-s + (−0.809 + 0.587i)20-s + (0.876 − 0.481i)25-s + (0.0388 − 0.119i)26-s + (−0.929 − 1.12i)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: $\chi_{500} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ 0.402 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8728521437\)
\(L(\frac12)\) \(\approx\) \(0.8728521437\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.187 + 0.982i)T \)
5 \( 1 + (-0.968 + 0.248i)T \)
good3 \( 1 + (-0.876 - 0.481i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
11 \( 1 + (0.929 - 0.368i)T^{2} \)
13 \( 1 + (-0.110 - 0.0604i)T + (0.535 + 0.844i)T^{2} \)
17 \( 1 + (1.62 + 0.645i)T + (0.728 + 0.684i)T^{2} \)
19 \( 1 + (-0.876 + 0.481i)T^{2} \)
23 \( 1 + (0.637 + 0.770i)T^{2} \)
29 \( 1 + (0.929 + 1.12i)T + (-0.187 + 0.982i)T^{2} \)
31 \( 1 + (-0.728 - 0.684i)T^{2} \)
37 \( 1 + (0.620 - 0.582i)T + (0.0627 - 0.998i)T^{2} \)
41 \( 1 + (-0.791 - 1.68i)T + (-0.637 + 0.770i)T^{2} \)
43 \( 1 + (-0.309 + 0.951i)T^{2} \)
47 \( 1 + (0.425 + 0.904i)T^{2} \)
53 \( 1 + (0.0800 + 1.27i)T + (-0.992 + 0.125i)T^{2} \)
59 \( 1 + (-0.968 + 0.248i)T^{2} \)
61 \( 1 + (0.824 - 1.75i)T + (-0.637 - 0.770i)T^{2} \)
67 \( 1 + (0.187 + 0.982i)T^{2} \)
71 \( 1 + (0.425 + 0.904i)T^{2} \)
73 \( 1 + (1.06 + 0.134i)T + (0.968 + 0.248i)T^{2} \)
79 \( 1 + (-0.876 - 0.481i)T^{2} \)
83 \( 1 + (-0.876 + 0.481i)T^{2} \)
89 \( 1 + (-0.371 - 0.0469i)T + (0.968 + 0.248i)T^{2} \)
97 \( 1 + (-1.26 - 1.52i)T + (-0.187 + 0.982i)T^{2} \)
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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

−10.94576269741290994494892502341, −10.05048305935103830081860199366, −9.451238998778888468831664853796, −8.639211580811035424943622371978, −7.51489584790299073975813462793, −6.30266157236778646685200718534, −4.98997812582305601836269883282, −4.26731288259609705501103061345, −2.64140822529207079351385635157, −1.63327058147142198140840081932, 1.79165070802897530418121007781, 3.76605411608165221841687885834, 4.89184283325386378248023349970, 5.97570181769522825788980521309, 6.71165694955873542378576322159, 7.44822456936258555405219173022, 8.865190315026639567546043121064, 9.237100457082718978420333042456, 10.30707328838272877553023789456, 10.90911006284317328178813652446

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