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Properties

Label 1-504-504.229-r1-0-0
Degree $1$
Conductor $504$
Sign $0.888 - 0.458i$
Analytic cond. $54.1623$
Root an. cond. $54.1623$
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  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s − 31-s + (0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s − 47-s + (0.5 − 0.866i)53-s − 55-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s − 31-s + (0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s − 47-s + (0.5 − 0.866i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(54.1623\)
Root analytic conductor: \(54.1623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (1:\ ),\ 0.888 - 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.434724208 - 0.3479216043i\)
\(L(\frac12)\) \(\approx\) \(1.434724208 - 0.3479216043i\)
\(L(1)\) \(\approx\) \(0.9684099665 + 0.04630832008i\)
\(L(1)\) \(\approx\) \(0.9684099665 + 0.04630832008i\)

Euler product

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

−23.72244618032306484716617077169, −22.70191328403087647099865131638, −21.61457516142343570072496230461, −21.0967204140521941760556788951, −19.96249309587793202846510288066, −19.36815648149386885994197913069, −18.588692140141625223727507221047, −17.32641877859844117094966431006, −16.44984758388290791343695633451, −16.20156701583446663384502137151, −14.70469354645660534606202537678, −14.22568919907358436220586125066, −12.87702188201019844221465390318, −12.30836235691958957339524702261, −11.41031312362811497721444418342, −10.41075122906870190341648550255, −9.21483210148029088955182848997, −8.53370559417027069938936614469, −7.66154620958864440588878104115, −6.42529306334206494847086408747, −5.49048936773762081690578396393, −4.29402215192827585763620441318, −3.6262303347224123918987744929, −2.01082091883212585940648837315, −0.8611123618412955770197249264, 0.49411377379015407599240244518, 2.17328166442888852513134163122, 3.13895103554057597235474787258, 4.208579176310814340814973384157, 5.29090902536838462126050599017, 6.55484305921043674532193362652, 7.33336541876738259451086868513, 8.09576643690447107598675956884, 9.511618191463335087175120767290, 10.14485175601296970260419871000, 11.27643180224212908974346941362, 11.92778727013062078909553096228, 12.94325214689335795339688824486, 14.0190992688691828780900395415, 14.94051060410019243254539142400, 15.40174398348330555287436991543, 16.50286077308080568078437145207, 17.674426066140001082824847909923, 18.085593361206000762299288953168, 19.32911838345250130732537525138, 19.789923417924182742436995354798, 20.75991768521964174119237986154, 21.922777099141369357586889666188, 22.52321333027805132559724868183, 23.236284248574760536758588010

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