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Properties

Label 1-532-532.339-r0-0-0
Degree $1$
Conductor $532$
Sign $0.881 + 0.471i$
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  + (−0.939 − 0.342i)3-s + (−0.766 + 0.642i)5-s + (0.766 + 0.642i)9-s + (0.5 − 0.866i)11-s + (−0.766 − 0.642i)13-s + (0.939 − 0.342i)15-s + (−0.766 + 0.642i)17-s + (−0.173 + 0.984i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (0.173 − 0.984i)29-s + 31-s + (−0.766 + 0.642i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)3-s + (−0.766 + 0.642i)5-s + (0.766 + 0.642i)9-s + (0.5 − 0.866i)11-s + (−0.766 − 0.642i)13-s + (0.939 − 0.342i)15-s + (−0.766 + 0.642i)17-s + (−0.173 + 0.984i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (0.173 − 0.984i)29-s + 31-s + (−0.766 + 0.642i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6934303983 + 0.1737303960i\)
\(L(\frac12)\) \(\approx\) \(0.6934303983 + 0.1737303960i\)
\(L(1)\) \(\approx\) \(0.6886826369 + 0.02070853192i\)
\(L(1)\) \(\approx\) \(0.6886826369 + 0.02070853192i\)

Euler product

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

−23.26993685853277125952920591869, −22.634304551759949218569017068468, −21.916346198264362389004541623474, −20.84301667827914718759516053119, −20.14133156665600343027049292863, −19.254924358177997839454916602051, −18.206302362504849118330771870043, −17.30444675568793075360661043122, −16.66804507810254671317810602541, −15.84517592506310499232705885885, −15.15480217370339054568324732059, −14.08207201688819453681017171158, −12.65888797526480213162981792983, −12.17251011824117741409954932674, −11.471508673539140165897556379736, −10.449410538312169567780898815246, −9.45922707704081697314341703255, −8.66652502997414777514847443680, −7.214014444910781041521217135636, −6.726262716320165175711178505619, −5.24512308677193596495059693150, −4.58260460684162895844132602578, −3.857047743468496739637750073039, −2.10196413257559149464229245135, −0.6104686611255179905814804824, 0.87393664795164432758038095096, 2.443237448912293630524468049460, 3.68819128294023166079351742786, 4.68969329465246418318842012178, 5.8933460103529960377599549357, 6.617468926224234829947473728176, 7.57921596143884284337109081935, 8.3711830772112062882348736808, 9.87071210632830750339965980796, 10.722956268808507099734595586452, 11.51536448169065036600532257969, 12.080241396049397091699280990559, 13.156698206781195544640081605016, 14.06814949287593151631056264768, 15.298796877596781048994690520526, 15.76822129756562086148319084201, 16.99630228870325920973201893625, 17.49326312382871870611220374477, 18.530082073940472256891812655775, 19.302378510305625411834034204943, 19.813122979826685890568605847151, 21.31089955811685374572580019575, 22.15381158596271633493327616627, 22.589871689047326140637178936601, 23.532591810079097956941599727183

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