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Properties

Label 1-143-143.112-r0-0-0
Degree $1$
Conductor $143$
Sign $-0.810 - 0.586i$
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  + (−0.587 + 0.809i)2-s + (0.309 − 0.951i)3-s + (−0.309 − 0.951i)4-s + (−0.587 − 0.809i)5-s + (0.587 + 0.809i)6-s + (−0.951 + 0.309i)7-s + (0.951 + 0.309i)8-s + (−0.809 − 0.587i)9-s + 10-s − 12-s + (0.309 − 0.951i)14-s + (−0.951 + 0.309i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (0.951 − 0.309i)18-s + (−0.951 − 0.309i)19-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (0.309 − 0.951i)3-s + (−0.309 − 0.951i)4-s + (−0.587 − 0.809i)5-s + (0.587 + 0.809i)6-s + (−0.951 + 0.309i)7-s + (0.951 + 0.309i)8-s + (−0.809 − 0.587i)9-s + 10-s − 12-s + (0.309 − 0.951i)14-s + (−0.951 + 0.309i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (0.951 − 0.309i)18-s + (−0.951 − 0.309i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.810 - 0.586i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ -0.810 - 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09757521122 - 0.3012420649i\)
\(L(\frac12)\) \(\approx\) \(0.09757521122 - 0.3012420649i\)
\(L(1)\) \(\approx\) \(0.5174869964 - 0.1348953962i\)
\(L(1)\) \(\approx\) \(0.5174869964 - 0.1348953962i\)

Euler product

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

−28.59403255123445290748040571557, −27.53257008861977688088209415365, −26.84557144425213103107484348760, −26.0817683361475910419581203313, −25.394505948688641959620129057621, −23.31633135838748866587585829301, −22.35989673541820283914486443096, −21.83095465596309525447034461096, −20.49413955809013729892865533453, −19.739535213761694108463726113332, −19.00647432603917024791259768855, −17.781991820527847962342590251945, −16.45599404356637437031767776558, −15.78543211411105166509694133550, −14.45708710999114105680151861047, −13.26956697091734468893692644584, −11.88816012369276421552680230742, −10.7976794499695229026122279223, −10.14166381859181357383215172733, −9.08267253956069000790575907132, −7.931814786505785036254307345861, −6.589161663604450842663855750110, −4.42113434572875213957823366926, −3.49038793897420267840732244331, −2.53869593337235290418670994329, 0.30641406678351243620473119166, 2.09051473378380360374223555783, 4.128143023609645284722466253719, 5.8426527753393669541583610863, 6.7008668043453594551966537759, 7.96737532880500121315462716518, 8.68191368639199093047235326982, 9.718610188929876290999001373220, 11.44207849386228386782264912492, 12.76361039150661953776336698466, 13.44315256929190877778351998329, 14.91468225528083766958771339547, 15.78361517881970787556488544565, 16.85847631113988279349545905403, 17.76208052099799924404839788243, 19.07771676555217459386699116581, 19.4468706265814932767325285076, 20.4489736342052763376508946539, 22.35059866378109609894031391804, 23.440861402179706036892076128349, 24.10590509363255487644451843576, 24.950252477614441977930330389995, 25.81310494352691985485810990889, 26.64111620238856464196777508302, 28.11471775990339040571511642167

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