Properties

Degree 1
Conductor 47
Sign $0.0296 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.203 − 0.979i)2-s + (0.682 − 0.730i)3-s + (−0.917 − 0.398i)4-s + (0.962 + 0.269i)5-s + (−0.576 − 0.816i)6-s + (−0.775 + 0.631i)7-s + (−0.576 + 0.816i)8-s + (−0.0682 − 0.997i)9-s + (0.460 − 0.887i)10-s + (−0.334 + 0.942i)11-s + (−0.917 + 0.398i)12-s + (−0.990 + 0.136i)13-s + (0.460 + 0.887i)14-s + (0.854 − 0.519i)15-s + (0.682 + 0.730i)16-s + (−0.334 − 0.942i)17-s + ⋯
L(s,χ)  = 1  + (0.203 − 0.979i)2-s + (0.682 − 0.730i)3-s + (−0.917 − 0.398i)4-s + (0.962 + 0.269i)5-s + (−0.576 − 0.816i)6-s + (−0.775 + 0.631i)7-s + (−0.576 + 0.816i)8-s + (−0.0682 − 0.997i)9-s + (0.460 − 0.887i)10-s + (−0.334 + 0.942i)11-s + (−0.917 + 0.398i)12-s + (−0.990 + 0.136i)13-s + (0.460 + 0.887i)14-s + (0.854 − 0.519i)15-s + (0.682 + 0.730i)16-s + (−0.334 − 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0296 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0296 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(47\)
\( \varepsilon \)  =  $0.0296 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{47} (25, \cdot )$
Sato-Tate  :  $\mu(23)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 47,\ (0:\ ),\ 0.0296 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7325044318 - 0.7545402156i$
$L(\frac12,\chi)$  $\approx$  $0.7325044318 - 0.7545402156i$
$L(\chi,1)$  $\approx$  0.9893038110 - 0.6886998446i
$L(1,\chi)$  $\approx$  0.9893038110 - 0.6886998446i

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

−34.04957098102760512974637187167, −32.94828456298864398013433836658, −32.41486194658608838642479399057, −31.47568522206174043310663005148, −29.899267190286780232493987189695, −28.45948949062004646309248533353, −26.72571511064285398618204776775, −26.33722882709472364846560917005, −25.11398898854940868806952511172, −24.18926579462292491506677288442, −22.40642057422020774626716968132, −21.70332958755505131005627804930, −20.325846195869297212361715008254, −18.791447052081855829105995807888, −17.01608421203720469325650018560, −16.397929823716583356211459158608, −14.9667618860153719008355973007, −13.80740320356456918336583421324, −13.01684532205419788440663595133, −10.24289215909095701261668372315, −9.31713894159671165257166745145, −7.95873813543737516415868282310, −6.21215442626191269096894183082, −4.789490575877882376781363850471, −3.16040985831181206234352890497, 2.037871138241561321859580388892, 3.009089610859768637736380604133, 5.34336438591355465742472792657, 7.09639844959235201723596721483, 9.23203921626474001137684469618, 9.81769853974332040826247935876, 11.90040397358835589585852564789, 12.96419177094793418825781075965, 13.8842423695235674801672167289, 15.13723311322744521997811697469, 17.62291029696487199482675594790, 18.43718312566399927034708619692, 19.5751829008931026140233408455, 20.62368426033586906915751669183, 21.89428695624010385583698880274, 22.91078394162421931993144752922, 24.55060868042220806503636919673, 25.62166765972686913118323349086, 26.73356969124632104687977805511, 28.621752182847639946073646656232, 29.206790735810270618298165351547, 30.24966087973127795631631125304, 31.38834473164760024912194758338, 32.14728576553807559558489794633, 33.5873961939908612141487535475

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