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Properties

Label 1-47-47.27-r0-0-0
Degree $1$
Conductor $47$
Sign $0.600 + 0.799i$
Analytic cond. $0.218267$
Root an. cond. $0.218267$
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  + (−0.990 + 0.136i)2-s + (0.854 + 0.519i)3-s + (0.962 − 0.269i)4-s + (−0.334 + 0.942i)5-s + (−0.917 − 0.398i)6-s + (−0.0682 − 0.997i)7-s + (−0.917 + 0.398i)8-s + (0.460 + 0.887i)9-s + (0.203 − 0.979i)10-s + (0.682 + 0.730i)11-s + (0.962 + 0.269i)12-s + (−0.576 + 0.816i)13-s + (0.203 + 0.979i)14-s + (−0.775 + 0.631i)15-s + (0.854 − 0.519i)16-s + (0.682 − 0.730i)17-s + ⋯
L(s)  = 1  + (−0.990 + 0.136i)2-s + (0.854 + 0.519i)3-s + (0.962 − 0.269i)4-s + (−0.334 + 0.942i)5-s + (−0.917 − 0.398i)6-s + (−0.0682 − 0.997i)7-s + (−0.917 + 0.398i)8-s + (0.460 + 0.887i)9-s + (0.203 − 0.979i)10-s + (0.682 + 0.730i)11-s + (0.962 + 0.269i)12-s + (−0.576 + 0.816i)13-s + (0.203 + 0.979i)14-s + (−0.775 + 0.631i)15-s + (0.854 − 0.519i)16-s + (0.682 − 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.600 + 0.799i$
Analytic conductor: \(0.218267\)
Root analytic conductor: \(0.218267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (0:\ ),\ 0.600 + 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6234525662 + 0.3115258406i\)
\(L(\frac12)\) \(\approx\) \(0.6234525662 + 0.3115258406i\)
\(L(1)\) \(\approx\) \(0.7824977724 + 0.2481416599i\)
\(L(1)\) \(\approx\) \(0.7824977724 + 0.2481416599i\)

Euler product

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

−34.4933725910660864334356000636, −32.47437850939068819905582785931, −31.656563406998100355936940068524, −30.267528754406058897015102250681, −29.25262144357966816126281198640, −27.934909062948030008439832279568, −27.11939953837282503256882364133, −25.57297451259322960366899909172, −24.85195254758152896999547033893, −24.00868573799488640640385228586, −21.65507325601032396920666083368, −20.47178335508733962558901714742, −19.490857670016343568344119465320, −18.67596574755327393629109258425, −17.23461435620308207547409580653, −15.90839622784319245313040185462, −14.69576638469379737343893532285, −12.66809464052774675917273891607, −11.94172009573656985723695142601, −9.80654029397465976921071332171, −8.59628621865952055529311624089, −7.96092007561691093432527316199, −6.03914891580440251031353973123, −3.32623649267657408511531242475, −1.56036451724587251375148228121, 2.36371859092513967007418106284, 4.06222512716185316106117853085, 6.886210709753775509260096032261, 7.70207314118078666955637143326, 9.43359320885263024097602787100, 10.27432970829019054405040063030, 11.62055785865565846061478326369, 14.04122654019958219092778512493, 14.93821880729305429043895152879, 16.19153415481042200215457886760, 17.4625080644349846839914976261, 19.03015535030772702149182416785, 19.747952021490490115827127839102, 20.84003663135223380708201515183, 22.39749095625134947776423602565, 23.985001512722880958355654466545, 25.44565847628876872593625211400, 26.326301261868936959284596182, 26.96561387828985011596060971077, 28.060026684939245341517417592927, 29.85226970938329816879232134817, 30.46185315074404805324402044431, 32.11801176828295068127557218718, 33.43371607823650382814174122716, 34.061964879296941700148271210394

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