Properties

Degree 1
Conductor 47
Sign $0.600 + 0.799i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.990 + 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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(47\)
\( \varepsilon \)  =  $0.600 + 0.799i$
motivic weight  =  \(0\)
character  :  $\chi_{47} (27, \cdot )$
Sato-Tate  :  $\mu(23)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 47,\ (0:\ ),\ 0.600 + 0.799i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6234525662 + 0.3115258406i$
$L(\frac12,\chi)$  $\approx$  $0.6234525662 + 0.3115258406i$
$L(\chi,1)$  $\approx$  0.7824977724 + 0.2481416599i
$L(1,\chi)$  $\approx$  0.7824977724 + 0.2481416599i

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

−34.4933725910660864334356000636, −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