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Properties

Label 1-143-143.87-r1-0-0
Degree $1$
Conductor $143$
Sign $0.964 - 0.265i$
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  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + 5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + 12-s + 14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s − 18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + 5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + 12-s + 14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s − 18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.173241064 - 0.2932036246i\)
\(L(\frac12)\) \(\approx\) \(2.173241064 - 0.2932036246i\)
\(L(1)\) \(\approx\) \(1.387188104 + 0.09065934219i\)
\(L(1)\) \(\approx\) \(1.387188104 + 0.09065934219i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−28.219250365935886178316051805, −27.543997841812467554291971381902, −26.33381704489277381103898387076, −25.06026016278203749286606518475, −23.93017642174821988528387094930, −22.73342858272168914530777407268, −21.935278812258355014800600523793, −21.20136339316377817226796708548, −20.68985309668284773565474132620, −19.14578585244806636030116745098, −18.00666656883676464877407811231, −17.24320946554926039072523215057, −15.69815003251227173721435566511, −14.71425853827939691052718435209, −13.825056099156016896626273361736, −12.38727837623974191226206508270, −11.61494333291644469198035976986, −10.36422926865357387115822727983, −9.72569784975790247524020922616, −8.593216824443647189014685563505, −5.985682590022424239958664940584, −5.54923988129579007237288778869, −4.28213118636342849476170472627, −2.85081537036189783082869045120, −1.43285909445314219198249874151, 0.85885065238182388284049915055, 2.65971998050514016960761674691, 4.6519553829591639819960612023, 5.60163522795627091169217809153, 6.74556614256071060802055592824, 7.50335847199809081692733137082, 8.820628245966214563593291673740, 10.389564040375101305141456945049, 11.77572211182199759678618682781, 12.90303850979914823450865590613, 13.8650880673157277127225165196, 14.279370728718291930628344853370, 16.08533383020355581049767473935, 17.02110858412512537300531987550, 17.73451621321837984548251462437, 18.45344148664628043636499567009, 20.17573081215198021717128640390, 21.314910755569483564169305823232, 22.3872659884579118131417223907, 23.18865480162638544804178322844, 24.207828394174372355938545863299, 24.78739269601077574862081038650, 25.78921071244943703990952133824, 26.73319515177086740016562303427, 28.05182367253346287227139565220

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