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Properties

Label 1-143-143.127-r1-0-0
Degree $1$
Conductor $143$
Sign $-0.803 - 0.595i$
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  + (−0.104 − 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (0.809 + 0.587i)5-s + (−0.104 + 0.994i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.5 − 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (−0.669 − 0.743i)15-s + (0.913 − 0.406i)16-s + (0.104 − 0.994i)17-s + (0.309 − 0.951i)18-s + (0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (0.809 + 0.587i)5-s + (−0.104 + 0.994i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.5 − 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (−0.669 − 0.743i)15-s + (0.913 − 0.406i)16-s + (0.104 − 0.994i)17-s + (0.309 − 0.951i)18-s + (0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.803 - 0.595i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ -0.803 - 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2356878637 - 0.7142231267i\)
\(L(\frac12)\) \(\approx\) \(0.2356878637 - 0.7142231267i\)
\(L(1)\) \(\approx\) \(0.5874195330 - 0.3451973102i\)
\(L(1)\) \(\approx\) \(0.5874195330 - 0.3451973102i\)

Euler product

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

−28.5150793256086059253226473204, −27.44098960864117876286539101493, −26.34218495544762788181048045606, −25.50106650268414861200026610241, −24.41395905503510390760873963615, −23.64948960562122358468760013149, −22.50664167397060599539442860805, −21.99265603144696478317524071509, −20.67087233871047280433247819697, −19.07135775828612596770303868861, −18.03988595122035959080798500840, −17.07490087318558388654981286208, −16.48344736535069895403558435185, −15.65864019668874655746476932837, −14.21887283243954512469770384206, −13.04275794600785588050224143946, −12.3420574395898962637743001284, −10.32590004072716210875115867428, −9.78167074370392390774965889060, −8.49896505897198992789990594731, −6.86300461632442817575008392302, −6.05280389825980041122491825309, −5.17632343331270295240059693385, −3.86553831615157778595307699718, −1.14947866160779757206547601696, 0.40296785545411421047411893318, 2.055541907084505106147377563696, 3.35122066193085926831743739222, 5.05730785879158317029320022516, 6.101942665876066421097797257328, 7.37340741435650106135336255275, 9.4189127700314547006208656957, 9.95525724100917388213520082399, 11.138077348147270685737536028711, 11.9971640866233432687259202396, 13.185799368495533028898483038992, 13.81548229448066352018173839303, 15.61787615869899979501557078858, 16.91115365701812377564948109876, 17.834051634421155045444024888599, 18.57399236089398173940898148573, 19.45801659487476626241195895728, 20.86170655404238001912132921267, 21.93093627485923855077495592157, 22.44442585834920737636312147483, 23.214771917779520579831405225962, 24.686491285588819814599278005587, 25.94673440512107420628269279137, 26.81951988968592710024277843109, 28.11170477758622671556899916923

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