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Properties

Label 1-143-143.37-r1-0-0
Degree $1$
Conductor $143$
Sign $-0.621 - 0.783i$
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  + (0.207 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.621 - 0.783i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ -0.621 - 0.783i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.305906038 - 2.702928183i\)
\(L(\frac12)\) \(\approx\) \(1.305906038 - 2.702928183i\)
\(L(1)\) \(\approx\) \(1.312483589 - 1.205486901i\)
\(L(1)\) \(\approx\) \(1.312483589 - 1.205486901i\)

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

−27.86999497321466273639680438577, −27.43944274175182434418098488882, −25.85647990337554519537769300581, −25.59209182266131640828858268714, −24.720360071912451603811199016753, −23.82866430285975185703867309900, −22.27965937795741551496748652234, −21.42389951210245484332267966460, −20.90967461827293471314830180392, −19.20311978044182308644180620536, −18.25427900321708606548506855149, −17.14173719334472822929748584344, −16.1376012476309356960440679727, −14.99362722837996130229732994852, −14.40993607709791302018736776064, −13.33808111242783329295603663678, −12.414306908755916487438494670144, −10.3366597765158173419595128260, −9.13306995022460023002473980820, −8.62854651426924738616914295316, −7.33595326190107642490123884385, −5.77349981238822331633102623725, −4.968339750843977287315157461559, −3.458357236148114164680956231488, −1.8978288830702788829475996148, 1.07996440501515800605285482918, 2.199277723166905917008826832139, 3.36493207627309071271661694777, 4.67548912610256160440030237444, 6.32690506572039038450051694880, 7.78530288052705433060253544356, 9.05720258043883714512397504880, 10.04033663071919178167262211897, 10.92691280279823141456674383961, 12.514254652519662308069296187983, 13.358556451314787549588799788854, 14.21610592087866342232564900317, 14.802305543586652101613429298293, 16.95102729358646101839447355075, 17.96501207284400792780696695996, 18.849699254082839719337526870575, 19.79149175104504841495882062106, 20.9558160447103029164696928297, 21.167876607013969722588816113113, 22.672538342975524736734850301567, 23.65069110941334815163859369655, 24.749225406354892675302661680101, 25.92412863685809639394800738791, 26.674672704906543751708345513950, 27.72578933530932389801072673599

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