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Properties

Label 1-532-532.87-r0-0-0
Degree $1$
Conductor $532$
Sign $0.182 + 0.983i$
Analytic cond. $2.47059$
Root an. cond. $2.47059$
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)3-s − 5-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)23-s + 25-s + 27-s + (−0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s − 33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s − 5-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)23-s + 25-s + 27-s + (−0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s − 33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.182 + 0.983i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ 0.182 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6774366929 + 0.5629967877i\)
\(L(\frac12)\) \(\approx\) \(0.6774366929 + 0.5629967877i\)
\(L(1)\) \(\approx\) \(0.7576490612 + 0.2549691406i\)
\(L(1)\) \(\approx\) \(0.7576490612 + 0.2549691406i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 - T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
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 \( 1 + T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 - T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 - T \)
83 \( 1 + T \)
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

−23.32520705963035955575358072730, −22.70411798069877036310657988358, −21.74184488414961145451180504599, −20.735285359221232134603786661114, −19.51739763600160824379997135014, −19.107931534699009269899041904671, −18.48361033178875445645649404437, −17.26556612774181506474338263483, −16.56896422483253502064629936208, −15.86211164834729130991068533228, −14.58445369329700455612146718573, −13.88711163528622154981694409560, −12.741940560019916469951330775844, −12.116864813218154786868539965799, −11.21438164120345596830257665901, −10.70538065503176735582001513729, −8.96681835482050723391735165135, −8.29302471422025571046563301023, −7.314398809060561328006339319760, −6.49285181867304655265689698325, −5.5945447016588681834389510690, −4.28037968179266258923091220214, −3.34487573225905274865415678201, −1.87017943382986150512452583323, −0.65279881248962232465775685666, 1.035434931536602280612561642479, 3.02496894518277592641321246471, 3.816635641734581739472867926579, 4.763997744824646390155350350970, 5.61543697897754984091578454905, 6.895309051508584575716302055960, 7.771960119733544514373661747201, 8.937606073829185283949276212393, 9.75642757304490780668889441428, 10.74747023827288879645001319497, 11.54622028604409918158823310253, 12.18637206904819932469145414466, 13.25954578797738749587767334371, 14.77458163986924367882425303953, 15.08836372055505065452833012511, 16.057802785959725418767233646053, 16.67468593440801061283139669651, 17.70903161624302851882145393638, 18.47215017013862540661611804596, 19.721585805497710298713151955202, 20.32351181564945494189278794678, 21.047357682320876020341923763464, 22.200096077401063051012250740207, 22.89022146101353770535620770939, 23.26230955508298551916113999614

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