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Properties

Label 1-143-143.97-r1-0-0
Degree $1$
Conductor $143$
Sign $0.738 + 0.674i$
Analytic cond. $15.3674$
Root an. cond. $15.3674$
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  + (0.994 + 0.104i)2-s + (−0.978 + 0.207i)3-s + (0.978 + 0.207i)4-s + (0.587 + 0.809i)5-s + (−0.994 + 0.104i)6-s + (0.207 − 0.978i)7-s + (0.951 + 0.309i)8-s + (0.913 − 0.406i)9-s + (0.5 + 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (−0.743 − 0.669i)15-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (0.951 − 0.309i)18-s + (0.743 − 0.669i)19-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (−0.978 + 0.207i)3-s + (0.978 + 0.207i)4-s + (0.587 + 0.809i)5-s + (−0.994 + 0.104i)6-s + (0.207 − 0.978i)7-s + (0.951 + 0.309i)8-s + (0.913 − 0.406i)9-s + (0.5 + 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (−0.743 − 0.669i)15-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (0.951 − 0.309i)18-s + (0.743 − 0.669i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.738 + 0.674i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.738 + 0.674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.813910013 + 1.091710121i\)
\(L(\frac12)\) \(\approx\) \(2.813910013 + 1.091710121i\)
\(L(1)\) \(\approx\) \(1.759511573 + 0.3947830909i\)
\(L(1)\) \(\approx\) \(1.759511573 + 0.3947830909i\)

Euler product

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

−28.384885560905861965354957443960, −27.32486287755151949426994647280, −25.36411241220747860864255896530, −24.71510010524090179006578409080, −24.03621639576631307439046368222, −22.85429390981516610940611719964, −22.07686129158514184611108214200, −21.21170917562736916556226163456, −20.37746003032321998841206643451, −18.821191422773974879298389989527, −17.826778350642400069798818581185, −16.503802341876368479743753580273, −15.96543479787112893079297146249, −14.55064002083596771559869340350, −13.3518799576450552892759043586, −12.39219893275750038488934274012, −11.82497541344986201369084723087, −10.57706651187429010634608291733, −9.24012721068998988283728078823, −7.501105054509121241395031101485, −6.07591334873422168780469149536, −5.40839235552381417860518150139, −4.5011729182471660793539226319, −2.49364309515557037523257631501, −1.14019727113423447680612225417, 1.43515472526922232012110088622, 3.2652584618381412145711541997, 4.490338276698517743390152211988, 5.66420450696413274019439179437, 6.6482526539004176562020316145, 7.506952315793692336428765229428, 9.91053400691672550957345276420, 10.80521892641502614535836123120, 11.51805744689085287261470427253, 12.91319278427998043252768700738, 13.7974675109795681750506907898, 14.871920668551767341281340232787, 15.91254152159014000800191115711, 17.10733907340604009956951248494, 17.69180239560154663224098076187, 19.25377332857024361884299402433, 20.58552900373403838969605977079, 21.6286134987394452697026968303, 22.171712130432159162009988511309, 23.30505504111596136600625820185, 23.730239857630746218258139040752, 25.04376393143963099120312649955, 26.19495096474846081158753984339, 27.05104518131460714091452745782, 28.58162145545574193312295251266

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