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Properties

Label 1-504-504.437-r0-0-0
Degree $1$
Conductor $504$
Sign $-0.220 - 0.975i$
Analytic cond. $2.34056$
Root an. cond. $2.34056$
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  − 5-s + 11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 23-s + 25-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯
L(s)  = 1  − 5-s + 11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 23-s + 25-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(2.34056\)
Root analytic conductor: \(2.34056\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (0:\ ),\ -0.220 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4538201149 - 0.5678575499i\)
\(L(\frac12)\) \(\approx\) \(0.4538201149 - 0.5678575499i\)
\(L(1)\) \(\approx\) \(0.7838949602 - 0.1546452928i\)
\(L(1)\) \(\approx\) \(0.7838949602 - 0.1546452928i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−24.02507543323054030421573475209, −23.06896528999927251877019318551, −22.15464633121008716002887466128, −21.53297886148983824664912102244, −20.29160403610967618295268788857, −19.484204722078294219625165123208, −19.172035802869800077557467932264, −17.87465061359561566418934673589, −16.99416141057207352601784550159, −16.22744775462954335976348051705, −15.22283437612344660306516994699, −14.61664522836475209769148613070, −13.5585824523658888987663609836, −12.44879149044597075623892288419, −11.70097319158777845430733122060, −11.03299720878773144499284030271, −9.7874633843496341085645542396, −8.82921193564031658626233034856, −8.00098981775085074987907004708, −6.91221751025142405806912362928, −6.196063358324348984992952514769, −4.5067263885292186503659724781, −4.13292044311945414142934724437, −2.78667252545436629920926158763, −1.41914793527186090298968510110, 0.406951823224147283550617622592, 2.048528389384035760798776623990, 3.4228450639380831991903913555, 4.17247683345892182629561271843, 5.299487308936458500022361055149, 6.51518399310869364083235649638, 7.461032476388318669573329030370, 8.26899080313373280064349838229, 9.26437281479118691989897613541, 10.31689568130073218849137944174, 11.33542789837076185354096871275, 12.07259867188743636346980051711, 12.81996260867916806644367202468, 14.103640047014541666574931332050, 14.83385723351525317988049836818, 15.691288073085755697625917457484, 16.50705156991045322505043893529, 17.3936796191109546404585613309, 18.405257235859586851051266134530, 19.24463686594522670207830285752, 20.0732750799288716174042244368, 20.55591820579254649633307983654, 22.0012353254391623166869980676, 22.527608054856740986990259648637, 23.30985054359240933938356640837

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