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Properties

Label 1-532-532.471-r0-0-0
Degree $1$
Conductor $532$
Sign $0.893 + 0.449i$
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 + (−0.939 + 0.342i)5-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (0.939 + 0.342i)13-s + (0.173 + 0.984i)15-s + (−0.939 + 0.342i)17-s + (−0.766 + 0.642i)23-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)27-s + (−0.766 + 0.642i)29-s + 31-s + (0.939 − 0.342i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)5-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (0.939 + 0.342i)13-s + (0.173 + 0.984i)15-s + (−0.939 + 0.342i)17-s + (−0.766 + 0.642i)23-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)27-s + (−0.766 + 0.642i)29-s + 31-s + (0.939 − 0.342i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.893 + 0.449i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ 0.893 + 0.449i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.013663760 + 0.2408561820i\)
\(L(\frac12)\) \(\approx\) \(1.013663760 + 0.2408561820i\)
\(L(1)\) \(\approx\) \(0.9323228017 - 0.06127398622i\)
\(L(1)\) \(\approx\) \(0.9323228017 - 0.06127398622i\)

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 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−23.201237394211991574246235446190, −22.57370465353559904996410373420, −21.740308937571197421576853396720, −20.77666949732527356269717912605, −20.15633649623667829394546265194, −19.42501687992239070542743573034, −18.46766626128724957379481940035, −17.26011440598066732141956174116, −16.35196253440345563089368788863, −15.83198497479044259552653238135, −15.11416268642873801919442433220, −14.0867656433300941045777984132, −13.24279844839076367387188506718, −11.93347442724051620686041406277, −11.22766934988924481500451003237, −10.54392899444699661620590464578, −9.24426596780005331588839988262, −8.60611315990055943539270525865, −7.85703650271366648955821520810, −6.40198317136906847825766878992, −5.415734720984757869407292437059, −4.17757693144490819754561489812, −3.78163386962604002341210218361, −2.54615701824801188685983751928, −0.614929207028378415435922536259, 1.2243590430619259303448353908, 2.3445970832786853038327765204, 3.57440734193926820744728235564, 4.45121710058398366665932143279, 6.0553893813499177234017301258, 6.7930191624574737395606991933, 7.62258367530556102544431317110, 8.44084293044416590615707163391, 9.36658993535502147447200338567, 10.81495604388932082878109104583, 11.57894781645143978165474589141, 12.28278390024351077402501140511, 13.20738602377637694751946133064, 14.088584979712354009272061321547, 14.99771794891838796485923206566, 15.73153823816820409969989893509, 16.87902111354853948862098647295, 17.87433033288662077457446186179, 18.45295059852542459723411179585, 19.431397624907365504319507419555, 19.91706607353482271351324563271, 20.7458236246826287369634043818, 22.14156269332108515251142344661, 22.84841431089123189862619407712, 23.612997710676465667607831798576

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