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Properties

Label 1-500-500.39-r1-0-0
Degree $1$
Conductor $500$
Sign $0.514 + 0.857i$
Analytic cond. $53.7324$
Root an. cond. $53.7324$
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)3-s + (−0.809 − 0.587i)7-s + (−0.929 + 0.368i)9-s + (−0.968 + 0.248i)11-s + (0.929 − 0.368i)13-s + (−0.876 − 0.481i)17-s + (0.187 − 0.982i)19-s + (−0.425 + 0.904i)21-s + (0.0627 − 0.998i)23-s + (0.535 + 0.844i)27-s + (−0.992 − 0.125i)29-s + (−0.876 − 0.481i)31-s + (0.425 + 0.904i)33-s + (−0.535 + 0.844i)37-s + (−0.535 − 0.844i)39-s + ⋯
L(s)  = 1  + (−0.187 − 0.982i)3-s + (−0.809 − 0.587i)7-s + (−0.929 + 0.368i)9-s + (−0.968 + 0.248i)11-s + (0.929 − 0.368i)13-s + (−0.876 − 0.481i)17-s + (0.187 − 0.982i)19-s + (−0.425 + 0.904i)21-s + (0.0627 − 0.998i)23-s + (0.535 + 0.844i)27-s + (−0.992 − 0.125i)29-s + (−0.876 − 0.481i)31-s + (0.425 + 0.904i)33-s + (−0.535 + 0.844i)37-s + (−0.535 − 0.844i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(53.7324\)
Root analytic conductor: \(53.7324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 500,\ (1:\ ),\ 0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2323620417 + 0.1315712118i\)
\(L(\frac12)\) \(\approx\) \(0.2323620417 + 0.1315712118i\)
\(L(1)\) \(\approx\) \(0.6530093902 - 0.2877328848i\)
\(L(1)\) \(\approx\) \(0.6530093902 - 0.2877328848i\)

Euler product

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

−23.1906629964030222311164976452, −22.36348155113427754612945039829, −21.633374996185078453215411834608, −20.925501180880195871826781241745, −20.09761524162459726103964566092, −19.02648504944830826084474648152, −18.26681995372679827957417195225, −17.213129300073031341550212560524, −16.153624324892349040667729474130, −15.80367024995970818938863531981, −14.98245672975046602069895161458, −13.81966572071786501175180216997, −12.93834817074301935176582987758, −11.9031893952800205174334075934, −10.89026061180831308625934168117, −10.26901955048832932448743406232, −9.12315941641880954273896520296, −8.64945370115415002146922434805, −7.22234063896127500725731441335, −5.82496305204194409896703755948, −5.532861270309622680234624456578, −3.987755013260100174755283732198, −3.35592378151140308010469852749, −2.04875130553429678036603756064, −0.08817732033224590634659595041, 0.839280609289969725906875817462, 2.29868268387832968552575419879, 3.206379494053764692825185425286, 4.63545202338690655584786990252, 5.81268308721125000818120092023, 6.697335485789949876118662740128, 7.4335381589522290197956754356, 8.413330400854454596734022427435, 9.46563334375264900302763426709, 10.71826548883745960846903161935, 11.28270083316350870892171879771, 12.62096909125060020751847531555, 13.214342607605721001610503056, 13.68050475101272466890085321031, 15.03516839086587204522894247393, 16.02038472081312259932954349456, 16.77998949829243099880631518502, 17.90208455332747190807046191170, 18.34886530687898629666240098590, 19.31724421005272628597857643079, 20.17441157205679476021249379337, 20.76237603351637337961039112368, 22.34448431781441183353866646597, 22.71139306534436559511353409659, 23.70495914868852852176974237785

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