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Properties

Label 1-143-143.73-r0-0-0
Degree $1$
Conductor $143$
Sign $0.261 - 0.965i$
Analytic cond. $0.664089$
Root an. cond. $0.664089$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.261 - 0.965i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.261 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.291608998 - 0.9880279079i\)
\(L(\frac12)\) \(\approx\) \(1.291608998 - 0.9880279079i\)
\(L(1)\) \(\approx\) \(1.378634663 - 0.6354035006i\)
\(L(1)\) \(\approx\) \(1.378634663 - 0.6354035006i\)

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

−28.677734426796633893161738223436, −27.830642419226645763310535683126, −26.110010942343611638215236537568, −25.58073686240540499766706129385, −24.35842740594771159475415965032, −23.52264690368291394756559074446, −22.30510002751261496611695962532, −21.75344558629007537381775222066, −21.136971998025033870287087681090, −19.82597405191523916088303574174, −18.11062989493251232877461769513, −17.129153490891322687298771447382, −16.26990583492391530275288926566, −15.379818544815872383933436586524, −14.32308128196753380130146969061, −12.900933203742137385135642845958, −12.36197748528637115262996095083, −11.03868044429100842563961820758, −9.91147253353423120399064553449, −8.65776857337058701042652086326, −6.66532139098793280263084536691, −5.890579473062477910082184095752, −5.091456968762139320280399159395, −3.738648057541985771248466811348, −2.1669266720892368256996871806, 1.36577911433103401255319718282, 2.755138030557480367474667133349, 4.38956017317881601594557590332, 5.73666038462512019932016398220, 6.457922893905631640152473803884, 7.47272510466119061703863225308, 9.890024411319433055082859363467, 10.54063425714662562058869080546, 11.74054909151327844618592227263, 12.8199462146849414785403040117, 13.60598328960670888951062393567, 14.39211742033396541010152148638, 16.09212749826268045507253942318, 16.85825517063730625641047650744, 18.105977453109207435261057740495, 19.11211964400006590676573406618, 20.27155920208145116503275723312, 21.39443459545755262221414038153, 22.35270560439843469698494335073, 23.00613548238853910778951476231, 23.87579021270069303139145854694, 24.99658818460525858823157860385, 25.71461204208448436421910083926, 27.33486427964164977693293316462, 28.667345774488559969858586112990

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