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Properties

Label 1-143-143.62-r1-0-0
Degree $1$
Conductor $143$
Sign $0.854 + 0.519i$
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  + (−0.978 − 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (−0.309 + 0.951i)5-s + (−0.978 + 0.207i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (0.5 − 0.866i)10-s + 12-s + (−0.809 − 0.587i)14-s + (0.104 + 0.994i)15-s + (0.669 + 0.743i)16-s + (0.978 − 0.207i)17-s + (−0.809 + 0.587i)18-s + (−0.104 + 0.994i)19-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (−0.309 + 0.951i)5-s + (−0.978 + 0.207i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (0.5 − 0.866i)10-s + 12-s + (−0.809 − 0.587i)14-s + (0.104 + 0.994i)15-s + (0.669 + 0.743i)16-s + (0.978 − 0.207i)17-s + (−0.809 + 0.587i)18-s + (−0.104 + 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.854 + 0.519i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.854 + 0.519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.629201384 + 0.4561271115i\)
\(L(\frac12)\) \(\approx\) \(1.629201384 + 0.4561271115i\)
\(L(1)\) \(\approx\) \(1.071370659 + 0.07180037116i\)
\(L(1)\) \(\approx\) \(1.071370659 + 0.07180037116i\)

Euler product

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

−27.79316022712426670846399545868, −26.99196096464719742241482323130, −26.18243666080525841079814966510, −25.06153203922712380431807221891, −24.37853813194900656614217765775, −23.496874202229389764213779789664, −21.42105956228556867502128315185, −20.73272740975678768303449873237, −19.96215351050361222078050730823, −19.15333039057916401744467925466, −17.84298679084230615322592292075, −16.78643924023290148185291889122, −15.938818802380255327532444583508, −14.9278621195768040821930681346, −13.94104287912513651462632403288, −12.41526532928545409276416781539, −11.09336308096616240952661081473, −10.003390253093162247649554175988, −8.878321581571802100384234948336, −8.17428036273147086846046047316, −7.29663410984086533173007975260, −5.3269288570378531456389765281, −4.057582143670404726644326443, −2.260902373119483275772224360025, −0.90205418266256702981263236121, 1.41877176225769340504349502680, 2.58438776069949459816473115275, 3.70670482958311421903890900537, 6.090226613494581203248770699888, 7.55678340398499958176276612580, 7.91601916918904977821540661341, 9.22640883563047830801892081592, 10.31882803439990165305683618764, 11.51025956461068643912653243226, 12.41868152935430775191741822726, 14.15835131866993688764433649476, 14.88198674373713730048003498067, 15.88871125854008374838486982373, 17.43520370452124620063954807805, 18.50580091225313264741782244621, 18.81046272148496713820529554395, 19.99348629421123201322607726075, 20.90657599225642315203888273775, 21.80737771150942719467713274689, 23.48926185914285807976953245810, 24.516470005439794106232111959950, 25.49783690368839621225354683930, 26.102548368006050878099550687710, 27.286178793232328155055298091680, 27.648970455542657699431514959964

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