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Properties

Label 1-143-143.128-r0-0-0
Degree $1$
Conductor $143$
Sign $0.434 + 0.900i$
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  + (−0.207 + 0.978i)2-s + (0.913 − 0.406i)3-s + (−0.913 − 0.406i)4-s + (0.951 + 0.309i)5-s + (0.207 + 0.978i)6-s + (−0.406 + 0.913i)7-s + (0.587 − 0.809i)8-s + (0.669 − 0.743i)9-s + (−0.5 + 0.866i)10-s − 12-s + (−0.809 − 0.587i)14-s + (0.994 − 0.104i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (0.587 + 0.809i)18-s + (0.994 + 0.104i)19-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (0.913 − 0.406i)3-s + (−0.913 − 0.406i)4-s + (0.951 + 0.309i)5-s + (0.207 + 0.978i)6-s + (−0.406 + 0.913i)7-s + (0.587 − 0.809i)8-s + (0.669 − 0.743i)9-s + (−0.5 + 0.866i)10-s − 12-s + (−0.809 − 0.587i)14-s + (0.994 − 0.104i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (0.587 + 0.809i)18-s + (0.994 + 0.104i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.434 + 0.900i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.434 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.140062816 + 0.7155542187i\)
\(L(\frac12)\) \(\approx\) \(1.140062816 + 0.7155542187i\)
\(L(1)\) \(\approx\) \(1.156100296 + 0.5055774135i\)
\(L(1)\) \(\approx\) \(1.156100296 + 0.5055774135i\)

Euler product

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

−28.240963646210353145198700253083, −26.997941057655853097562562892527, −26.37006012662906522017257695858, −25.506058932065132459923367705944, −24.3344809387386926960462081560, −22.783651475035890909316124005477, −21.84596617318729952276847343076, −20.96430025067784783712512727649, −20.18453924855144851611898354228, −19.51370708275362668406406677508, −18.21298202770117901276448581225, −17.203463342239814198986963096742, −16.11936976356202139537056988891, −14.498419286372019402236413365690, −13.32665650860927715621997657193, −13.27424919120957206240055841043, −11.36744478962915698951129498931, −10.11138757925871020067797328604, −9.585515818102748794408224489647, −8.56722041542487723028538866588, −7.18274833514507836441843878175, −5.14225013376839579007401097855, −3.94648600449030490307073587230, −2.783831102713450412174833307306, −1.48740413369813207247651329646, 1.82279149630850601399151430257, 3.2577478116501233589641817640, 5.14112145238937495590368952026, 6.3577185519683641941623332915, 7.17739276908315843629318304569, 8.724933462221219094925525452647, 9.18381569458704404835318231700, 10.372925397689188216642373745803, 12.51422681478951470705468241237, 13.41378001889767814792232897781, 14.319518086984263197426286782980, 15.15971370829887164220947971627, 16.17234965627782545015370433367, 17.62313295500708134924906223451, 18.35852121006085916598775433857, 19.09520700818390511413693397509, 20.4085701448543151157634606891, 21.790567182564569132589753956703, 22.47290631433394861830146467245, 24.02252757086003344047955153400, 24.79623912395483725526820577165, 25.460757567190928668449275292325, 26.18477923788875946184184166729, 27.06158955932304646023313217344, 28.522396858998126102411439552522

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