Properties

Degree 1
Conductor 47
Sign $0.00342 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(47\)
\( \varepsilon \)  =  $0.00342 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{47} (13, \cdot )$
Sato-Tate  :  $\mu(46)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 47,\ (1:\ ),\ 0.00342 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5112255252 - 0.5129798991i$
$L(\frac12,\chi)$  $\approx$  $0.5112255252 - 0.5129798991i$
$L(\chi,1)$  $\approx$  0.7207814736 - 0.07413702400i
$L(1,\chi)$  $\approx$  0.7207814736 - 0.07413702400i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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