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Properties

Label 1-47-47.13-r1-0-0
Degree $1$
Conductor $47$
Sign $0.00342 - 0.999i$
Analytic cond. $5.05085$
Root an. cond. $5.05085$
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  + (−0.334 + 0.942i)2-s + (0.203 − 0.979i)3-s + (−0.775 − 0.631i)4-s + (0.0682 + 0.997i)5-s + (0.854 + 0.519i)6-s + (−0.576 − 0.816i)7-s + (0.854 − 0.519i)8-s + (−0.917 − 0.398i)9-s + (−0.962 − 0.269i)10-s + (−0.460 − 0.887i)11-s + (−0.775 + 0.631i)12-s + (−0.682 − 0.730i)13-s + (0.962 − 0.269i)14-s + (0.990 + 0.136i)15-s + (0.203 + 0.979i)16-s + (0.460 − 0.887i)17-s + ⋯
L(s)  = 1  + (−0.334 + 0.942i)2-s + (0.203 − 0.979i)3-s + (−0.775 − 0.631i)4-s + (0.0682 + 0.997i)5-s + (0.854 + 0.519i)6-s + (−0.576 − 0.816i)7-s + (0.854 − 0.519i)8-s + (−0.917 − 0.398i)9-s + (−0.962 − 0.269i)10-s + (−0.460 − 0.887i)11-s + (−0.775 + 0.631i)12-s + (−0.682 − 0.730i)13-s + (0.962 − 0.269i)14-s + (0.990 + 0.136i)15-s + (0.203 + 0.979i)16-s + (0.460 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.00342 - 0.999i$
Analytic conductor: \(5.05085\)
Root analytic conductor: \(5.05085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (1:\ ),\ 0.00342 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5112255252 - 0.5129798991i\)
\(L(\frac12)\) \(\approx\) \(0.5112255252 - 0.5129798991i\)
\(L(1)\) \(\approx\) \(0.7207814736 - 0.07413702400i\)
\(L(1)\) \(\approx\) \(0.7207814736 - 0.07413702400i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (-0.334 + 0.942i)T \)
3 \( 1 + (0.203 - 0.979i)T \)
5 \( 1 + (0.0682 + 0.997i)T \)
7 \( 1 + (-0.576 - 0.816i)T \)
11 \( 1 + (-0.460 - 0.887i)T \)
13 \( 1 + (-0.682 - 0.730i)T \)
17 \( 1 + (0.460 - 0.887i)T \)
19 \( 1 + (0.0682 - 0.997i)T \)
23 \( 1 + (0.334 + 0.942i)T \)
29 \( 1 + (-0.682 + 0.730i)T \)
31 \( 1 + (-0.203 - 0.979i)T \)
37 \( 1 + (0.962 + 0.269i)T \)
41 \( 1 + (-0.854 - 0.519i)T \)
43 \( 1 + (0.775 + 0.631i)T \)
53 \( 1 + (0.854 + 0.519i)T \)
59 \( 1 + (-0.775 + 0.631i)T \)
61 \( 1 + (0.962 - 0.269i)T \)
67 \( 1 + (0.576 - 0.816i)T \)
71 \( 1 + (-0.334 - 0.942i)T \)
73 \( 1 + (0.917 - 0.398i)T \)
79 \( 1 + (-0.990 - 0.136i)T \)
83 \( 1 + (0.460 + 0.887i)T \)
89 \( 1 + (-0.0682 - 0.997i)T \)
97 \( 1 + (0.203 - 0.979i)T \)
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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

−34.08624426938371373977583508640, −32.494586043676924869641614457020, −31.7543500082566136521787736245, −30.8760303356671004519803305332, −28.842038487735578743040404416335, −28.493262642221711797698961549567, −27.41182181385636860491044673765, −26.18147925436510848069086642027, −25.12021698896025453042792028592, −23.11214887243879382827799685167, −21.85376862021384427736362761987, −21.013620330771930889003970785486, −20.05496227208771461412215748353, −18.883731548369477727429158448376, −17.17100814363967066444468045506, −16.23500526937755633770360330317, −14.66074390748963159771247400730, −12.860923768219952941004898228879, −11.95745500345811131378718704460, −10.169079552928959094373792383376, −9.35707945922057366617126212828, −8.25226044793790325538495908855, −5.28613800400254259189317738568, −4.024590165014081591207689955353, −2.24087163137718549623073921964, 0.45984567522448413171395804049, 3.089906704886672584401810120004, 5.69540850183203570520634103562, 7.037118991779392741187685978571, 7.70737113861237291413284923073, 9.53142726175145847411160783060, 11.01849897239731616743764791781, 13.1978988173697519568856364752, 13.96316265180131064052285979369, 15.23561449430988963221831856964, 16.807270715851664141777029488080, 17.968691467682343402058373067232, 18.909015963809700832186936025815, 19.87993576960159633619044717815, 22.24491668338077545303618098927, 23.224509746474363997155444675729, 24.177445593763341101597355538646, 25.48428351998209918131959118568, 26.23662320195752934810110629938, 27.27469353898938894395766136550, 29.191684350514632345362757582796, 29.8733980804565949012937509562, 31.36761999168637493154010584785, 32.41023284522964143665078397151, 33.85603886903175865861188336115

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