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Properties

Label 1-23e2-529.461-r0-0-0
Degree $1$
Conductor $529$
Sign $0.732 + 0.680i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
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  + (−0.917 − 0.398i)2-s + (−0.917 − 0.398i)3-s + (0.682 + 0.730i)4-s + (−0.576 + 0.816i)5-s + (0.682 + 0.730i)6-s + (0.682 − 0.730i)7-s + (−0.334 − 0.942i)8-s + (0.682 + 0.730i)9-s + (0.854 − 0.519i)10-s + (0.460 + 0.887i)11-s + (−0.334 − 0.942i)12-s + (0.854 + 0.519i)13-s + (−0.917 + 0.398i)14-s + (0.854 − 0.519i)15-s + (−0.0682 + 0.997i)16-s + (0.854 + 0.519i)17-s + ⋯
L(s)  = 1  + (−0.917 − 0.398i)2-s + (−0.917 − 0.398i)3-s + (0.682 + 0.730i)4-s + (−0.576 + 0.816i)5-s + (0.682 + 0.730i)6-s + (0.682 − 0.730i)7-s + (−0.334 − 0.942i)8-s + (0.682 + 0.730i)9-s + (0.854 − 0.519i)10-s + (0.460 + 0.887i)11-s + (−0.334 − 0.942i)12-s + (0.854 + 0.519i)13-s + (−0.917 + 0.398i)14-s + (0.854 − 0.519i)15-s + (−0.0682 + 0.997i)16-s + (0.854 + 0.519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.732 + 0.680i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.732 + 0.680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5555548058 + 0.2181298537i\)
\(L(\frac12)\) \(\approx\) \(0.5555548058 + 0.2181298537i\)
\(L(1)\) \(\approx\) \(0.5660850959 + 0.0007470985316i\)
\(L(1)\) \(\approx\) \(0.5660850959 + 0.0007470985316i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.917 - 0.398i)T \)
3 \( 1 + (-0.917 - 0.398i)T \)
5 \( 1 + (-0.576 + 0.816i)T \)
7 \( 1 + (0.682 - 0.730i)T \)
11 \( 1 + (0.460 + 0.887i)T \)
13 \( 1 + (0.854 + 0.519i)T \)
17 \( 1 + (0.854 + 0.519i)T \)
19 \( 1 + (-0.917 - 0.398i)T \)
29 \( 1 + (0.460 + 0.887i)T \)
31 \( 1 + (-0.990 + 0.136i)T \)
37 \( 1 + (-0.0682 - 0.997i)T \)
41 \( 1 + (0.962 + 0.269i)T \)
43 \( 1 + (-0.775 + 0.631i)T \)
47 \( 1 + (-0.990 - 0.136i)T \)
53 \( 1 + (0.854 + 0.519i)T \)
59 \( 1 + (-0.917 + 0.398i)T \)
61 \( 1 + (-0.576 + 0.816i)T \)
67 \( 1 + (0.460 + 0.887i)T \)
71 \( 1 + (0.962 - 0.269i)T \)
73 \( 1 + (0.460 - 0.887i)T \)
79 \( 1 + (-0.775 + 0.631i)T \)
83 \( 1 + (-0.576 + 0.816i)T \)
89 \( 1 + (0.203 + 0.979i)T \)
97 \( 1 + (0.962 - 0.269i)T \)
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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

−23.50622763826945359405858265852, −22.85078452322056336995931759619, −21.40219799601137782408939665549, −20.96413941084653894183143486445, −19.98290341768584008881000595291, −18.79638832962873509299453240893, −18.38025894232143856061762255332, −17.21049479892846766702045091336, −16.74229485065248303588257423910, −15.873373987242483556993087504759, −15.34876501884289612147843019853, −14.31194291526163750193632689135, −12.77137009878765565368691598561, −11.71769549341923815369163846013, −11.35728090105465095072841767585, −10.35262153137661008793231953436, −9.23972076786184271301389441895, −8.4895514250285504448262338960, −7.77484891213130981556678848106, −6.32932899418890692481103969823, −5.65293337594864337984225907932, −4.83228099350578896290234587274, −3.498488124077343785225463676000, −1.61483991086362656366821041031, −0.59655398144533965425940240641, 1.1478063441059525198673981900, 2.034857723095422349597339703760, 3.64549598144556956687744188960, 4.48010079393160644297807026765, 6.1785642715127964459099489579, 7.01603181094306049956977423188, 7.55360647543818728966205052035, 8.56987971644014027146827161647, 9.96737583952016511269433087609, 10.8114914301613013291594934861, 11.17964088098094618703936171240, 12.1044085130002358056654638856, 12.89409641010242559186202854853, 14.26725196628512731012147803612, 15.21883764694641984711967304125, 16.37191664143071753850030001437, 16.92243838546419157691719601000, 18.01887961760119477777549700179, 18.19019163584931639764838122907, 19.371844303605978476295085488995, 19.86010702938208908218432472651, 21.14690678751991728730178733657, 21.7278422030274912009642906429, 22.991254753044513254325278343748, 23.40019298671106388693354851524

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