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Properties

Label 1-143-143.81-r0-0-0
Degree $1$
Conductor $143$
Sign $0.569 - 0.821i$
Analytic cond. $0.664089$
Root an. cond. $0.664089$
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  + (−0.978 − 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (0.309 − 0.951i)5-s + (−0.978 + 0.207i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (−0.5 + 0.866i)10-s + 12-s + (−0.809 − 0.587i)14-s + (−0.104 − 0.994i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.809 + 0.587i)18-s + (−0.104 + 0.994i)19-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (0.309 − 0.951i)5-s + (−0.978 + 0.207i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (−0.5 + 0.866i)10-s + 12-s + (−0.809 − 0.587i)14-s + (−0.104 − 0.994i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.809 + 0.587i)18-s + (−0.104 + 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.569 - 0.821i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.569 - 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9582638625 - 0.5017189850i\)
\(L(\frac12)\) \(\approx\) \(0.9582638625 - 0.5017189850i\)
\(L(1)\) \(\approx\) \(0.9739791604 - 0.3209877685i\)
\(L(1)\) \(\approx\) \(0.9739791604 - 0.3209877685i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.978 - 0.207i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−28.09527627986780602852679181972, −27.195571944106227483181730161879, −26.39638124641834529633545014753, −25.92436460755207627080730823887, −24.72981984817537012756520411013, −23.97704398869603605679430478261, −22.28861729336208077093586931239, −21.21559084122587127490281507061, −20.30399767441366045094313221712, −19.4531792034387541011921810439, −18.32560905180040544329406727118, −17.63696242599460180751140077080, −16.30617405299580715168390835904, −15.12228791462357852395364749965, −14.561614421114916200251682819286, −13.45372305982448789783650403043, −11.34572168992592553749044928999, −10.63380820025073849649208939606, −9.61753564764235306530234604529, −8.514071920797826496940477672380, −7.53203465344792963191444827185, −6.521566240772776599298167442109, −4.66474492603721414423492167546, −2.91113298008250831832407786812, −1.87326742432711465243894170547, 1.440641715066081773855425400747, 2.273186206653290942052512443300, 4.05424039608559974233924021111, 5.87540648308775186960588773784, 7.45216907664782097265095154206, 8.40865467819631807481894855062, 8.99976728472106933536029731680, 10.151610547253041952093962002356, 11.695622211049001878770223396219, 12.57228179007561942961950699355, 13.7763500977420462936232611962, 15.08089507141423922918388478083, 16.05458617193517680957461131659, 17.434174857523689697674641104716, 18.03812870005045708784962780408, 19.22900264376852358647308708495, 20.08315285913682280975383494886, 20.93713577365220126368113123674, 21.540826372873087110384860003362, 23.7489409255119697796898878758, 24.72599338499788142557468969781, 25.019301329380723122974494332057, 26.23026775419642305917352321681, 27.17406825671983719481108473449, 28.069680531452958915002070802873

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