Properties

Label 1-47-47.19-r1-0-0
Degree $1$
Conductor $47$
Sign $0.257 - 0.966i$
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  + (−0.775 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.257 - 0.966i$
Analytic conductor: \(5.05085\)
Root analytic conductor: \(5.05085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (1:\ ),\ 0.257 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7119474009 - 0.5472392682i\)
\(L(\frac12)\) \(\approx\) \(0.7119474009 - 0.5472392682i\)
\(L(1)\) \(\approx\) \(0.6621830170 - 0.2839815916i\)
\(L(1)\) \(\approx\) \(0.6621830170 - 0.2839815916i\)

Euler product

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

−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