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Properties

Label 1-47-47.41-r1-0-0
Degree $1$
Conductor $47$
Sign $-0.473 + 0.880i$
Analytic cond. $5.05085$
Root an. cond. $5.05085$
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  + (0.682 − 0.730i)2-s + (−0.990 − 0.136i)3-s + (−0.0682 − 0.997i)4-s + (−0.460 + 0.887i)5-s + (−0.775 + 0.631i)6-s + (−0.917 + 0.398i)7-s + (−0.775 − 0.631i)8-s + (0.962 + 0.269i)9-s + (0.334 + 0.942i)10-s + (−0.203 + 0.979i)11-s + (−0.0682 + 0.997i)12-s + (−0.854 − 0.519i)13-s + (−0.334 + 0.942i)14-s + (0.576 − 0.816i)15-s + (−0.990 + 0.136i)16-s + (0.203 + 0.979i)17-s + ⋯
L(s)  = 1  + (0.682 − 0.730i)2-s + (−0.990 − 0.136i)3-s + (−0.0682 − 0.997i)4-s + (−0.460 + 0.887i)5-s + (−0.775 + 0.631i)6-s + (−0.917 + 0.398i)7-s + (−0.775 − 0.631i)8-s + (0.962 + 0.269i)9-s + (0.334 + 0.942i)10-s + (−0.203 + 0.979i)11-s + (−0.0682 + 0.997i)12-s + (−0.854 − 0.519i)13-s + (−0.334 + 0.942i)14-s + (0.576 − 0.816i)15-s + (−0.990 + 0.136i)16-s + (0.203 + 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $-0.473 + 0.880i$
Analytic conductor: \(5.05085\)
Root analytic conductor: \(5.05085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (1:\ ),\ -0.473 + 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09090031094 + 0.1520642683i\)
\(L(\frac12)\) \(\approx\) \(0.09090031094 + 0.1520642683i\)
\(L(1)\) \(\approx\) \(0.6483900853 - 0.1637882619i\)
\(L(1)\) \(\approx\) \(0.6483900853 - 0.1637882619i\)

Euler product

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

−33.47535230349158465643256313684, −32.13295946646907090865616287656, −31.75957366673770050612613846508, −29.762173025006334310619932498906, −29.04672391964313977806618455716, −27.453754576063209954295303791384, −26.49960931533020101273658372360, −24.82124243812283740447406132892, −23.87920387755040108636537834435, −23.019582079423965081922287429292, −21.94514118530183719135093982951, −20.747427800553631529879976693867, −18.98433718238383381932305882545, −17.13473187654977259904745040162, −16.44308588265338272635670759821, −15.6639971545157199079290665755, −13.71861324706424442490631327846, −12.5351588726623900661589113609, −11.59827593835403683414897967769, −9.57692696491010360657690204468, −7.7501584457690818506984709563, −6.32195389624856860725825111047, −5.06963202173982454217726326472, −3.7595494339243556303102447294, −0.09037887307542898642511897305, 2.49889865474376245639814594794, 4.27274887575579122360871816428, 5.89583269931243232047475303230, 7.06723543688232798632296661106, 9.89527878290563771103510178979, 10.76827732433700900320767425882, 12.17662127085474880749385357371, 12.86071788522796846114162075013, 14.811634787833235320145240005771, 15.74922962635524499411981594971, 17.65945267572827467497284750106, 18.86289512897531407120589094649, 19.78313357462792453099583052279, 21.648949247579389937750618331706, 22.49488303343360710616426698876, 23.113237800512321266535604753188, 24.37010926002189665642852721394, 26.13528042846900489673922835988, 27.76638773782971598294415250732, 28.48354629276440470597874853153, 29.739450338503376248794104810553, 30.41979394702782578727956683195, 31.77792552441328241974070714157, 32.95624215456207344926660660319, 34.14621164426757029617279257021

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