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Properties

Label 1-143-143.140-r1-0-0
Degree $1$
Conductor $143$
Sign $-0.841 + 0.540i$
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  + (0.669 + 0.743i)2-s + (−0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (−0.309 − 0.951i)5-s + (0.669 − 0.743i)6-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.5 − 0.866i)10-s + 12-s + (−0.809 + 0.587i)14-s + (−0.913 + 0.406i)15-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (−0.809 − 0.587i)18-s + (0.913 + 0.406i)19-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (−0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (−0.309 − 0.951i)5-s + (0.669 − 0.743i)6-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.5 − 0.866i)10-s + 12-s + (−0.809 + 0.587i)14-s + (−0.913 + 0.406i)15-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (−0.809 − 0.587i)18-s + (0.913 + 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.841 + 0.540i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ -0.841 + 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2688009297 + 0.9147581399i\)
\(L(\frac12)\) \(\approx\) \(0.2688009297 + 0.9147581399i\)
\(L(1)\) \(\approx\) \(0.9830675615 + 0.3233505853i\)
\(L(1)\) \(\approx\) \(0.9830675615 + 0.3233505853i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.669 + 0.743i)T \)
3 \( 1 + (-0.104 - 0.994i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (-0.104 + 0.994i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.913 + 0.406i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.913 + 0.406i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−27.65226827482767593565341201702, −26.75363510257202075897430219784, −26.14493089373820224898909150733, −24.38355910180886772523794031390, −23.20537933398097268391853559892, −22.51136758582699770370866898202, −21.93882934683405612239588119926, −20.5362519727027097225071206662, −20.120166902474698812420506274361, −18.864278995588218382366683299056, −17.67581471861179019802730034025, −16.200099578306047086090128697039, −15.26763611632366225368790379736, −14.28776920187429025235632893576, −13.52589274715046289864982135764, −11.81652440617070357032608506798, −10.99321912086228524911687075059, −10.276214046157514990982187478020, −9.267482161500238269824565992417, −7.29700027573006472304477728453, −5.98351559914737906332328141265, −4.545449412306641392183600679142, −3.70346682625781276243219448399, −2.60982754672863099694123826398, −0.28317715510551826719383644078, 1.84711440417369159310719873576, 3.5029368480719583222694772598, 5.181199599897408348606242316822, 5.896938407008474942721877033377, 7.23399210950028704354019975196, 8.29486411873791397533568415503, 9.06433514156026507604850797279, 11.526123403454739292822514107645, 12.30716370964788708540060851088, 12.99690164680568231115670479176, 14.03961552014453934879197488428, 15.32006213607362196654491749699, 16.21906306074693168357058635540, 17.31672269234023615960499196585, 18.18312461256030071199145850923, 19.449275222879658164656294058241, 20.5446177140381714312651308755, 21.78534544619400926973569036232, 22.72524820050539308549258559049, 23.86935943566630305852604179090, 24.36968969594391601591171684564, 25.16119266897235810491678561038, 26.03255657594247387754851739422, 27.5490141308252314196230120950, 28.58244757806177551600147400996

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