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Properties

Label 1-500-500.27-r0-0-0
Degree $1$
Conductor $500$
Sign $0.514 + 0.857i$
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  + (0.982 − 0.187i)3-s + (−0.587 + 0.809i)7-s + (0.929 − 0.368i)9-s + (−0.968 + 0.248i)11-s + (0.368 + 0.929i)13-s + (−0.481 + 0.876i)17-s + (−0.187 + 0.982i)19-s + (−0.425 + 0.904i)21-s + (0.998 + 0.0627i)23-s + (0.844 − 0.535i)27-s + (0.992 + 0.125i)29-s + (−0.876 − 0.481i)31-s + (−0.904 + 0.425i)33-s + (0.844 + 0.535i)37-s + (0.535 + 0.844i)39-s + ⋯
L(s)  = 1  + (0.982 − 0.187i)3-s + (−0.587 + 0.809i)7-s + (0.929 − 0.368i)9-s + (−0.968 + 0.248i)11-s + (0.368 + 0.929i)13-s + (−0.481 + 0.876i)17-s + (−0.187 + 0.982i)19-s + (−0.425 + 0.904i)21-s + (0.998 + 0.0627i)23-s + (0.844 − 0.535i)27-s + (0.992 + 0.125i)29-s + (−0.876 − 0.481i)31-s + (−0.904 + 0.425i)33-s + (0.844 + 0.535i)37-s + (0.535 + 0.844i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(2.32199\)
Root analytic conductor: \(2.32199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 500,\ (0:\ ),\ 0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.430710459 + 0.8101164350i\)
\(L(\frac12)\) \(\approx\) \(1.430710459 + 0.8101164350i\)
\(L(1)\) \(\approx\) \(1.298076675 + 0.2514355093i\)
\(L(1)\) \(\approx\) \(1.298076675 + 0.2514355093i\)

Euler product

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

−23.51688644390360163644572223253, −22.73531017632396769609002627768, −21.679700362910327915575457204377, −20.83897325144779512086727124364, −20.109574283919338382872751145591, −19.532079223636932543661070617889, −18.5088493681337391997984603971, −17.70849501033298823926666202599, −16.437157028870716008142672375932, −15.73463809294603020975120707614, −15.05855164219880051540317156941, −13.803121610576140294117823713078, −13.35031743863918452133519942570, −12.617713847276901705348893234, −10.94223044125590070957058040642, −10.42055136787590685015003543245, −9.36954365156771651241004595579, −8.550779319430942585610561760978, −7.51216815029668364868456707313, −6.85084652018746048072614633123, −5.33782806517803864703521178136, −4.31840246065984445129000883615, −3.16962760452988519806297284097, −2.56339216220301609261828048101, −0.80055818891235998063724549668, 1.65477928788132018416641083703, 2.56894187820719908501341833276, 3.53853724639217631450757311927, 4.65407964494330852357947190498, 6.01303048766038198429978410731, 6.88318136459325787008446050434, 8.04392695752941351456279804903, 8.74431385901190660972858548908, 9.60157221518506554089886458989, 10.49688339334538525405846768515, 11.78870615142797031010679335817, 12.85487598822688123250528981312, 13.23792619879356424672537202804, 14.453233563151878499476344472267, 15.17288735944847948865382045147, 15.89847348575071309705795203168, 16.8587314543638092998457886386, 18.35125048874514455147914983921, 18.62756285086781530655695827285, 19.53243519606368918009718164421, 20.37050486453252703694820575014, 21.37634421911147509511195672204, 21.73468270877956641047932580434, 23.18432804557431644677230495756, 23.76370372988943547499121988896

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