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Properties

Label 1-143-143.47-r1-0-0
Degree $1$
Conductor $143$
Sign $-0.590 - 0.806i$
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  + (0.951 + 0.309i)2-s + (−0.809 + 0.587i)3-s + (0.809 + 0.587i)4-s + (−0.951 + 0.309i)5-s + (−0.951 + 0.309i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (0.309 − 0.951i)9-s − 10-s − 12-s + (−0.809 + 0.587i)14-s + (0.587 − 0.809i)15-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (0.587 − 0.809i)18-s + (−0.587 − 0.809i)19-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (−0.809 + 0.587i)3-s + (0.809 + 0.587i)4-s + (−0.951 + 0.309i)5-s + (−0.951 + 0.309i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (0.309 − 0.951i)9-s − 10-s − 12-s + (−0.809 + 0.587i)14-s + (0.587 − 0.809i)15-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (0.587 − 0.809i)18-s + (−0.587 − 0.809i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.590 - 0.806i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ -0.590 - 0.806i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1993563983 + 0.3930139501i\)
\(L(\frac12)\) \(\approx\) \(-0.1993563983 + 0.3930139501i\)
\(L(1)\) \(\approx\) \(0.7818432197 + 0.4968862975i\)
\(L(1)\) \(\approx\) \(0.7818432197 + 0.4968862975i\)

Euler product

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

−27.90417507462927205908603922215, −26.45109681181030723619260673508, −25.01473035421025837614314608020, −23.98753093934115445768089815399, −23.41452381526061244273008428983, −22.7293079200197118192606512989, −21.732717001747665324567590410413, −20.312094046083410425926316065320, −19.55352981250991137817951281360, −18.70156586357359520986185680685, −16.98046012195591775351536399575, −16.284217205450603706222135205554, −15.19488662886351974249671744981, −13.79006939627520247536162303893, −12.79481727746599205781989609989, −12.1530022855366545016894316542, −11.04826981459873230670406443992, −10.20831755643664620013510499035, −8.0064041743320085401504413262, −6.92205351044417963547576865940, −5.95189404110553439435728631115, −4.53270952273903870175593327181, −3.61285151372512566666480146333, −1.652896603819097786763135275992, −0.132298823252725700312912313544, 2.76125168562223986680387242642, 3.95848840017915829259293644877, 4.999968352046178610839110293583, 6.20485096428819276691671225235, 7.11753493461266646910440447928, 8.69035032715394292825868357610, 10.29277956032687598233736889901, 11.61555705877408924680201413323, 11.9594176953446227238461351834, 13.237351355508480476429334290, 14.78734142926956696813368473615, 15.647192636602270282334745271912, 16.05527557749359568589644074796, 17.32466509457949868544103822809, 18.667719006135496912639644499524, 19.94104423282356896094014428716, 21.06397096005467338704025238478, 22.23937865439560320826100579533, 22.52465080669411239222747266918, 23.58290166456303534266479020551, 24.40244545518106395109988759992, 25.75609492872720610954329585332, 26.6427911581484576761437500594, 27.81995140755657245534213427815, 28.697463718514342944583669170424

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