Properties

Degree 1
Conductor 47
Sign $0.172 - 0.984i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.172 - 0.984i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.172 - 0.984i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(47\)
\( \varepsilon \)  =  $0.172 - 0.984i$
motivic weight  =  \(0\)
character  :  $\chi_{47} (14, \cdot )$
Sato-Tate  :  $\mu(23)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 47,\ (0:\ ),\ 0.172 - 0.984i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4758733766 - 0.3995980024i$
$L(\frac12,\chi)$  $\approx$  $0.4758733766 - 0.3995980024i$
$L(\chi,1)$  $\approx$  0.6595110087 - 0.3384315762i
$L(1,\chi)$  $\approx$  0.6595110087 - 0.3384315762i

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

−34.06910474321238195107950188758, −33.44292619724987455838835084620, −32.44078095204756557476005793409, −31.12689370490951666564833441167, −28.928801804185320025726641653303, −28.49168687275717220326100415808, −27.48084082168815526248302562590, −26.100074593027858427579036748842, −25.3892761562601228529216520434, −24.16451000448601991552740233100, −22.45410487538720709425558231833, −20.982996364183161197726466366808, −20.40005144457920208796876511795, −18.42888924449421061450077537666, −17.5808949383407726923336969532, −16.163926869540962156845100629377, −15.4834637909056920032297022903, −13.9737347009728221298744526135, −11.82434263907849222313147893004, −10.33605591156971470996450888422, −9.33208250572717027269300443499, −8.36008103973472930643944386949, −6.088395677356789608427278316023, −5.03350746365763971695024939329, −2.27948753028496174110266255538, 1.40416399807169883613068814858, 3.02875171508629861099487657365, 6.11614896734522005203129793376, 7.328550721279540014929256027939, 8.58316515524739853557247644254, 10.39040272251937255027961917294, 11.23790362529465025450793087533, 13.04454680269858472427635651325, 13.90456657035807027368746257722, 16.118141209018498378320195181964, 17.68780188031344069446494887864, 17.99302312356538110603194209757, 19.37535802351956766883527380577, 20.48937297360312740053093059191, 21.81104575348294965942619189774, 23.45673606144041990159647562261, 24.69462973438456810844102500618, 25.92898095646677435195582523145, 26.57954460498040738769701960220, 28.38624230120758804102642251953, 29.24635542543397440272798515485, 30.11159933540880723499810768569, 30.86950640342951430353495146041, 33.030579139886048768521292269866, 34.1336507530646085584095800725

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