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Properties

Label 1-143-143.8-r0-0-0
Degree $1$
Conductor $143$
Sign $0.753 + 0.657i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.753 + 0.657i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.753 + 0.657i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4400114666 + 0.1651219269i\)
\(L(\frac12)\) \(\approx\) \(0.4400114666 + 0.1651219269i\)
\(L(1)\) \(\approx\) \(0.5178731278 + 0.06627431118i\)
\(L(1)\) \(\approx\) \(0.5178731278 + 0.06627431118i\)

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

−28.04120248957927848872168736756, −27.517203936615034008661118475822, −26.407348443521650219846637908, −24.92166616970107024203510751417, −24.42270940228662171916795167915, −23.56343145930060635425546547262, −22.51015843538633470162647614260, −21.04896129820662040317579036590, −19.83150260404620916018501848947, −18.92940271792914958167185086764, −18.119593032970965526028497728040, −17.28670450741386062756356716180, −16.02956102671995833004298069609, −15.547159585469691285710958976836, −13.99671928769729502695919046654, −12.114032814722025731645188937389, −11.76752600710534724120575071597, −10.68049764266811150642984484402, −9.1431120541320999196876076166, −7.99943212187914671573033615830, −7.256723892972117457355839377000, −5.889110673223137511075494065823, −4.81312835198557912241241914816, −2.41173899673415020770394661435, −0.78290185800664870127659789994, 1.152522027534157389097841668484, 3.4033308607555479655726216649, 4.40296764731940860137855291109, 6.24502861163529183365975003580, 7.46692728764230722524359307833, 8.39819994406826356075231798978, 10.04728582076037380668048045315, 10.65506922219411421206268173653, 11.63728857883778531337834566046, 12.3876373612967415672434941251, 14.44031487560478808564461976587, 15.642169792581980216701543307884, 16.410425477191490712892947057667, 17.37276274711905515193031681949, 18.23003058308851761424474151489, 19.407954487738171461153267631746, 20.38362060682394348695489067234, 21.259038202264872807817001289417, 22.4268225845765364155799893272, 23.506374821256843344932592122904, 24.31477898365766705241669708398, 26.00361124346935459489408090055, 26.757422718505122861169219260516, 27.44689355237233465092309609118, 28.109254368000968663576745246678

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