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  :  $\chi_{47} (37, \cdot )$
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

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

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