Properties

Label 1-47-47.14-r0-0-0
Degree $1$
Conductor $47$
Sign $0.172 - 0.984i$
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  + (−0.917 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.172 - 0.984i$
Analytic conductor: \(0.218267\)
Root analytic conductor: \(0.218267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (0:\ ),\ 0.172 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4758733766 - 0.3995980024i\)
\(L(\frac12)\) \(\approx\) \(0.4758733766 - 0.3995980024i\)
\(L(1)\) \(\approx\) \(0.6595110087 - 0.3384315762i\)
\(L(1)\) \(\approx\) \(0.6595110087 - 0.3384315762i\)

Euler product

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

−34.06910474321238195107950188758, −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