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Properties

Label 1-532-532.367-r0-0-0
Degree $1$
Conductor $532$
Sign $-0.282 + 0.959i$
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  + (0.173 − 0.984i)3-s + (−0.173 + 0.984i)5-s + (−0.939 − 0.342i)9-s − 11-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)15-s + (0.939 − 0.342i)17-s + (−0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s + (−0.173 + 0.984i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.173 + 0.984i)5-s + (−0.939 − 0.342i)9-s − 11-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)15-s + (0.939 − 0.342i)17-s + (−0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s + (−0.173 + 0.984i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.282 + 0.959i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ -0.282 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3341112515 + 0.4465675039i\)
\(L(\frac12)\) \(\approx\) \(0.3341112515 + 0.4465675039i\)
\(L(1)\) \(\approx\) \(0.7947170086 + 0.01511877040i\)
\(L(1)\) \(\approx\) \(0.7947170086 + 0.01511877040i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−23.13635615548265790460149767337, −22.35276593815778812489640632557, −21.26433358826759352237996135421, −20.856836724049290284156455321481, −20.02260767485382428438846143147, −19.29702355374600577834225105617, −18.02427277730117673096652591027, −17.02735371258203868832154833824, −16.417808251596566939449164958668, −15.6090033051494643969865758790, −14.89858138247790372127319521088, −13.87258589561899328547795611940, −12.82335530237517467392223217027, −12.092576635542165144164992925044, −10.93642388149731859799210477501, −10.015614174025489264671002634819, −9.425317978844418978524365930742, −8.13892493445321917077485352527, −7.84896247089335969117175957207, −5.92124629582151855726424451455, −5.17570466450450859736605556296, −4.392282028716073497450170566885, −3.31741021882467739505425727710, −2.15638278361179038067459316672, −0.26940712178690825956965508514, 1.64060623077694257378902548756, 2.6809393632264763830460624496, 3.45427027636875505786533133205, 5.09010675982363277446748597387, 6.08947910860738039541827854366, 7.21181226071302751814837430543, 7.52471060468110072647389515946, 8.65760574311611899703458891196, 9.89445648707640939501355630105, 10.762945326352292959480078353777, 11.84758575599282050036722381223, 12.39540292695141024757704047572, 13.635221072605690932997150463340, 14.18304556877127292927375595191, 15.00985469065378805243942605659, 16.04863394843515792181003128531, 17.14978104761329480687511460571, 18.11677724001504562513346123291, 18.59501141006419770091123458710, 19.35550268084467620033566887558, 20.13905680881977224174807036056, 21.269198356073669358969126850175, 22.11612288798838908642390583846, 23.102624608861365442168850401425, 23.67445059497828356596994465622

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