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Properties

Label 1-143-143.125-r1-0-0
Degree $1$
Conductor $143$
Sign $-0.0439 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0439 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0439 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.0439 + 0.999i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ -0.0439 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6030281898 + 0.6301198051i\)
\(L(\frac12)\) \(\approx\) \(0.6030281898 + 0.6301198051i\)
\(L(1)\) \(\approx\) \(0.6554356846 + 0.1655945132i\)
\(L(1)\) \(\approx\) \(0.6554356846 + 0.1655945132i\)

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

−27.84193416148050731957131940173, −26.92183139273932931815241143649, −26.21828472349966417470687676238, −24.93720795394404242567430509958, −24.04810297268173558054166560471, −22.63463478364822113060668793073, −21.58004338063174235180256826412, −20.741402713383103986853752726781, −20.11200182809155796757345906140, −18.215797432460128360385168884337, −17.85676467151338856011511595824, −16.69400284096234057804791746015, −16.2869469020096097485179090675, −14.69110202811750068960356310239, −13.26798921757554189021617496071, −11.9150284450274839523721402057, −11.005278772229573161761037669757, −10.00135162265947356184183161867, −9.35422854504853018049441863952, −7.819115491820514524363244408782, −6.52054200166846218822940076422, −5.28316692419045598841060918684, −3.81249755807397695066653404051, −1.86874005978836914630600586010, −0.52524816686750850569916208444, 1.42435370015518509842090149936, 2.3525021082629669655646390199, 5.25632971092834227883177459388, 6.01242956789348949217307105564, 7.04504011751636420741448999450, 8.2914378064237194636941979519, 9.49367370958892972457938115908, 10.69221026612034960783329403824, 11.4868408968552190975414198468, 12.74420324944730926050694834290, 14.144501670333467695094774054907, 15.28676418357092309206707512560, 16.52325648513174407080852451117, 17.53315038198437075884076247908, 18.07910324648910341007414734351, 18.80597974346246007560672828622, 20.08973235260731701429235613456, 21.51817725853659004687775594858, 22.227611475590799127771387591425, 23.85054706098660669300310855370, 24.40573190420205976679787818403, 25.36901458329472294942833927853, 26.20592293350886365181892161978, 27.57355481909862308674415442677, 28.33508244442482684683268133152

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