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Properties

Label 1-143-143.35-r1-0-0
Degree $1$
Conductor $143$
Sign $-0.794 - 0.607i$
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  + (0.104 − 0.994i)2-s + (−0.978 + 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.809 + 0.587i)5-s + (0.104 + 0.994i)6-s + (0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (0.5 + 0.866i)10-s + 12-s + (0.309 − 0.951i)14-s + (0.669 − 0.743i)15-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (−0.309 − 0.951i)18-s + (−0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (−0.978 + 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.809 + 0.587i)5-s + (0.104 + 0.994i)6-s + (0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (0.5 + 0.866i)10-s + 12-s + (0.309 − 0.951i)14-s + (0.669 − 0.743i)15-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (−0.309 − 0.951i)18-s + (−0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.794 - 0.607i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ -0.794 - 0.607i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2024083461 - 0.5984549954i\)
\(L(\frac12)\) \(\approx\) \(0.2024083461 - 0.5984549954i\)
\(L(1)\) \(\approx\) \(0.5942667573 - 0.2776135795i\)
\(L(1)\) \(\approx\) \(0.5942667573 - 0.2776135795i\)

Euler product

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

−27.91102281508037298679521363699, −27.5724023377857298887510079811, −26.71437398312669621127582553961, −25.10632557259669596794332263054, −24.33702692601852632291960276518, −23.54695263167907689590004775022, −22.992631495520489449457377086671, −21.77556790042911853361266003891, −20.6575949575221124755375096225, −19.07568073224946241991686681562, −18.089676518287613785091941186792, −17.18466959724252906950096042332, −16.36780962084589612166227105021, −15.51596181726915263339969656207, −14.316555632643139443645105729644, −13.04312219137904415617257096380, −12.0437577942868040846304503417, −11.062728023377230521965443912991, −9.46170399190478994447475817543, −7.981279700105218486235206260926, −7.42123779307790014488951897671, −5.93880635326135536608822656036, −4.8994760537071729800697714982, −4.079044013518757565646061617376, −1.13057446022674330998220596779, 0.32784307116565901699952282073, 2.02711788895459594674019200809, 3.81833552282148152818707899939, 4.67773409582115518233389793142, 5.99299588639834335786695634210, 7.62391620192878682479147921330, 8.944785909029651221294655414084, 10.60751003722221912861975373497, 10.96230657242322765584862731074, 11.963851486763387622630549062917, 12.81342269746567342436356561187, 14.48995016995780595999830401804, 15.2412790407004990856113880843, 16.778569288240461457800372228391, 17.89402437980393817766468909208, 18.55833330498431616714393247705, 19.65309741532187872785382676567, 20.85303837745564409358717227648, 21.820910690678560997349595914018, 22.46175520597928936470745840785, 23.65940624378432630715394992715, 24.02185852798908436795006351750, 26.182329969932572084687612730802, 27.10196762414896361511812331732, 27.87894816590190014926885118769

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