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Properties

Label 1-143-143.129-r1-0-0
Degree $1$
Conductor $143$
Sign $0.530 + 0.847i$
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  + (0.309 − 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 10-s + 12-s + (−0.809 + 0.587i)14-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 10-s + 12-s + (−0.809 + 0.587i)14-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.530 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1306830444 + 0.07241491096i\)
\(L(\frac12)\) \(\approx\) \(0.1306830444 + 0.07241491096i\)
\(L(1)\) \(\approx\) \(0.5237271712 - 0.3274647246i\)
\(L(1)\) \(\approx\) \(0.5237271712 - 0.3274647246i\)

Euler product

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

−27.828492641589487760955978901727, −26.72795186314646307759324441607, −25.72634595924408506982372308637, −24.960263105395905931379276370755, −23.723279946660407199746327552181, −23.13219984400068934097517624054, −22.179865932771270880761337449378, −21.65442128226436022990957941934, −19.32371454754670997414703215844, −18.73982021425492382075586739512, −17.694177896349616896399510953155, −16.80117787882228889515337923533, −15.62642769350255460597827793745, −14.91138969318552023439880305488, −13.452308703834402920300134382996, −12.67653967601210367771442117105, −11.54271755164392064680586214208, −10.25405639936239077136833356139, −8.68284756282232328026783077793, −7.36758127948836475060794799855, −6.50371291602102460787548891705, −5.80076757216193416826483519936, −4.2364227758913307857773322498, −2.668892565654190650330968863821, −0.069266447936949241847383257418, 1.08974519970853279256000106276, 3.28836518476046724293561602272, 4.399469797144225069514464767592, 5.23652248710850141217067638605, 6.661157134159330587167645655, 8.719045930616089289606247006197, 9.70756786774637167839529791935, 10.62725655636466641968456050888, 11.74429191134332930166633611527, 12.604452600402011484759664144, 13.48498496538353704860501769874, 15.035352597544263428501620218291, 16.24040977988834628165935118863, 17.00469354882354760034909716111, 18.24145580805315622297628037458, 19.52696724204820574318101988015, 20.37531831084231045874727070359, 21.203545441213311147727726626703, 22.22331392192481730806750634386, 23.260372583816819467418026915023, 23.61196873071069017764837073953, 25.18795042420261010172104260395, 26.94344157914545745304905945472, 27.26768120899851875982255795862, 28.52322020346310148050202709775

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