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Properties

Label 1-143-143.61-r1-0-0
Degree $1$
Conductor $143$
Sign $-0.372 + 0.927i$
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  + (−0.913 − 0.406i)2-s + (0.669 − 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (−0.913 + 0.406i)6-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 + 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (−0.978 + 0.207i)15-s + (−0.104 + 0.994i)16-s + (−0.913 + 0.406i)17-s + (−0.309 + 0.951i)18-s + (0.978 + 0.207i)19-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (0.669 − 0.743i)3-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (−0.913 + 0.406i)6-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 + 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (−0.978 + 0.207i)15-s + (−0.104 + 0.994i)16-s + (−0.913 + 0.406i)17-s + (−0.309 + 0.951i)18-s + (0.978 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.372 + 0.927i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ -0.372 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1656539256 - 0.2450949142i\)
\(L(\frac12)\) \(\approx\) \(-0.1656539256 - 0.2450949142i\)
\(L(1)\) \(\approx\) \(0.4679148743 - 0.3595344160i\)
\(L(1)\) \(\approx\) \(0.4679148743 - 0.3595344160i\)

Euler product

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

−28.3478232753613589667426373099, −27.52258342036599304397345125811, −26.697091518465652099209452727039, −25.9958767585185941062310409160, −25.15473966887224943945445288060, −24.09052090406169042179227300995, −22.70862354355645576417876142238, −21.83922464624350519187765087822, −20.355227962696384423587493401424, −19.62628568561838544097633037308, −18.85827034455069496974077409056, −17.810295931801263282454546591790, −16.15561924640457537400800369483, −15.748252566539798049298110940836, −14.94889569671075545464460520294, −13.794662614709698840912247033723, −11.88865174389891777681685730334, −10.872719522950378481814799559238, −9.742368885069601609375284726136, −8.90996061232100929955083590984, −7.83882236726174565201032725127, −6.744211264384661501904758758098, −5.23272623459933700771804823601, −3.48557919232351166400010611431, −2.35909018039718093466548907004, 0.14474086673747474697139881420, 1.36278039512412277325454828606, 3.00027659338646854726219401068, 4.06907336622763016307628437557, 6.5597500577777464619224010245, 7.47925575578207069787131855821, 8.41238699043106403917187224272, 9.32947867058761458393525407836, 10.63171109908073742888397573501, 12.00322567765258829871153917464, 12.70604843333091403152724183750, 13.80981189150919946292460664215, 15.45564298250944673306426707252, 16.339356918252488416746258924788, 17.45425342969935736326326796133, 18.54596823662098998285843576720, 19.55549805186038833188790113345, 20.023002026376706351085481021645, 20.78697860938063578029706250705, 22.425102507922940348259171097465, 23.75260351083187799794283441230, 24.53364460241902690427020277074, 25.58069266593611068947106055515, 26.54499707634987307798728938713, 27.12690940132494897805443240586

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