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Properties

Label 1-500-500.47-r0-0-0
Degree $1$
Conductor $500$
Sign $0.816 + 0.577i$
Analytic cond. $2.32199$
Root an. cond. $2.32199$
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  + (−0.248 + 0.968i)3-s + (0.951 − 0.309i)7-s + (−0.876 − 0.481i)9-s + (0.187 + 0.982i)11-s + (0.481 − 0.876i)13-s + (0.368 − 0.929i)17-s + (0.968 − 0.248i)19-s + (0.0627 + 0.998i)21-s + (−0.904 − 0.425i)23-s + (0.684 − 0.728i)27-s + (0.637 + 0.770i)29-s + (0.929 + 0.368i)31-s + (−0.998 − 0.0627i)33-s + (0.684 + 0.728i)37-s + (0.728 + 0.684i)39-s + ⋯
L(s)  = 1  + (−0.248 + 0.968i)3-s + (0.951 − 0.309i)7-s + (−0.876 − 0.481i)9-s + (0.187 + 0.982i)11-s + (0.481 − 0.876i)13-s + (0.368 − 0.929i)17-s + (0.968 − 0.248i)19-s + (0.0627 + 0.998i)21-s + (−0.904 − 0.425i)23-s + (0.684 − 0.728i)27-s + (0.637 + 0.770i)29-s + (0.929 + 0.368i)31-s + (−0.998 − 0.0627i)33-s + (0.684 + 0.728i)37-s + (0.728 + 0.684i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(2.32199\)
Root analytic conductor: \(2.32199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 500,\ (0:\ ),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.357871334 + 0.4317857566i\)
\(L(\frac12)\) \(\approx\) \(1.357871334 + 0.4317857566i\)
\(L(1)\) \(\approx\) \(1.101009451 + 0.2574760684i\)
\(L(1)\) \(\approx\) \(1.101009451 + 0.2574760684i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.248 + 0.968i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (0.187 + 0.982i)T \)
13 \( 1 + (0.481 - 0.876i)T \)
17 \( 1 + (0.368 - 0.929i)T \)
19 \( 1 + (0.968 - 0.248i)T \)
23 \( 1 + (-0.904 - 0.425i)T \)
29 \( 1 + (0.637 + 0.770i)T \)
31 \( 1 + (0.929 + 0.368i)T \)
37 \( 1 + (0.684 + 0.728i)T \)
41 \( 1 + (-0.425 - 0.904i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (-0.844 - 0.535i)T \)
53 \( 1 + (0.998 - 0.0627i)T \)
59 \( 1 + (-0.992 + 0.125i)T \)
61 \( 1 + (-0.425 + 0.904i)T \)
67 \( 1 + (0.770 + 0.637i)T \)
71 \( 1 + (-0.535 + 0.844i)T \)
73 \( 1 + (-0.125 + 0.992i)T \)
79 \( 1 + (0.968 + 0.248i)T \)
83 \( 1 + (0.248 + 0.968i)T \)
89 \( 1 + (0.992 + 0.125i)T \)
97 \( 1 + (0.770 - 0.637i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.70921470791938037185259683347, −22.97036120935699715693422783074, −21.73497139216448317143651047343, −21.25840349494648304797291010150, −20.02426941646946753515184503757, −19.19110655859151378316344018382, −18.458524052765567737009832026209, −17.78124189113383958726803145475, −16.86442350634300276869513007832, −16.062754361640076827210662619976, −14.75073217604181889969439383081, −13.95163795758883345001022840809, −13.377396102275230964845319612940, −12.0072653960392165825036316349, −11.6434416130817409313711433792, −10.75481628883983952007594976693, −9.35305804576367615319910245000, −8.1537150491224180382670029828, −7.88907322385749472048563410300, −6.362242711457465923012882329732, −5.88442685478312797784938281020, −4.65451292545553657684235753746, −3.31719938311090847371804668269, −1.97461826342937994440330461125, −1.13255652491471560584860798688, 1.072333248098101478901275133851, 2.67406310400396604951392293221, 3.820471171739764699635152921703, 4.83606680631592862993145542965, 5.39802062013072013589185150376, 6.77995329212694364496693426012, 7.88928232133599844381995113508, 8.78918217451405243879709980447, 9.96419129959505640806001390145, 10.4337452923534994075030223003, 11.59502367251341994144865679588, 12.12004944536377977870093460945, 13.62940353857492000539912527453, 14.391425519530607986743663898451, 15.25411914069908485699321473856, 15.9579440476838504215500486913, 16.9193445486137089191005046855, 17.82343200464642346505235707870, 18.25708550958098545935303105883, 20.11447863948485471677138641491, 20.26722592364754927460103315355, 21.13707393159143603745011851543, 22.10403381353974600071698003984, 22.84243611441874366200358758200, 23.50286338093905342048718163437

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