Properties

Degree 1
Conductor 47
Sign $0.257 - 0.966i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.257 - 0.966i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.257 - 0.966i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(47\)
\( \varepsilon \)  =  $0.257 - 0.966i$
motivic weight  =  \(0\)
character  :  $\chi_{47} (19, \cdot )$
Sato-Tate  :  $\mu(46)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 47,\ (1:\ ),\ 0.257 - 0.966i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7119474009 - 0.5472392682i$
$L(\frac12,\chi)$  $\approx$  $0.7119474009 - 0.5472392682i$
$L(\chi,1)$  $\approx$  0.6621830170 - 0.2839815916i
$L(1,\chi)$  $\approx$  0.6621830170 - 0.2839815916i

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

−33.70510670747325349444328280116, −33.30270608341269507180299419127, −32.49601715754612781931313172770, −30.16724472509079376826805822648, −28.92270870698472610153727518732, −28.41346686318670144099528747920, −26.86265972888502302662847467840, −26.1620545123939675571055347990, −24.79114728940129105455861987188, −23.52152256400787678143651714275, −22.50520050622650416944775875213, −21.05084438837216958134581316658, −19.565602734792812374996971125699, −17.94413648637506070128917852536, −17.227530920181389262764871033825, −16.37187408440467493882365941331, −14.88374485787726805594970869759, −13.457363156625815482453917512606, −11.38868854949616517169423266941, −10.09725884914863748636452995242, −9.38410290948921838037064257851, −7.04513995003520820199670395741, −6.20760988772993563967990766500, −4.589751193886981910519953469933, −1.33918886916812134367981629721, 0.93074188185579695415500997551, 2.70321789050866430666063869838, 5.40162230031221707102558049351, 6.74521773403395485885751121869, 8.66165744374932321921424559406, 9.92407728651130714032215390921, 11.24247098303091091563090464785, 12.40393625216750541148858609002, 13.48386284223550251475975726597, 15.935752205664906275629475952297, 17.05277379190883697427664901183, 18.09342701855829046195195928613, 18.840912207670605782334728724883, 20.47357274369922027756009466074, 21.93039797680624517580712002023, 22.38002520453082212964476354547, 24.68800822603515605232893915611, 25.17072495388685793284261620188, 26.91194552033555803189929018457, 28.04881106295857632725664058898, 29.02044439617754724804299622593, 29.56529774181141130937728571303, 30.82772858708787769206979228681, 32.51494007833220662812625043269, 33.9180955437720423674712941625

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