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Properties

Label 1-143-143.10-r1-0-0
Degree $1$
Conductor $143$
Sign $0.0128 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.0128 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4162592472 + 0.4109552095i\)
\(L(\frac12)\) \(\approx\) \(0.4162592472 + 0.4109552095i\)
\(L(1)\) \(\approx\) \(0.4583640103 + 0.2568565444i\)
\(L(1)\) \(\approx\) \(0.4583640103 + 0.2568565444i\)

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

−27.84904773719679724477105516231, −27.30490127381978871654428582542, −25.84205382776910743680689443489, −24.98065438069880579120391897826, −23.67093434571794082790440637472, −22.71859467948186770388547288203, −22.0357433524108125524307237001, −20.54763907707706995875618561891, −19.58062543687902227623686787323, −18.67711864424829706955584273213, −18.31072131939940163621449516501, −16.788565092014773282381924735817, −16.03214221676137049855373654904, −14.33185460385236343239040807851, −12.82926020128367593342801624994, −12.22321031320157263791140386333, −11.497072584713359028643564984993, −10.313389036865603631234726414457, −8.790291768758791159044299704988, −7.91699366744822198246536776242, −6.74459583012658223137844696547, −5.108037893616518856963713702138, −3.46725999697231257184048358238, −2.18768557956673608464857178504, −0.534879966594984989281152619998, 0.64925539404779397959055572275, 3.69177318414987103142354691693, 4.55923016992403728373798605449, 5.97077208280998557845809167796, 7.11360167311370805356862681239, 8.24815711296620869166448777401, 9.513275780342678040028427169385, 10.48099838984100877185588289026, 11.40742019507134554432624665110, 13.00806463664089875106958657019, 14.50799560089412720562764467028, 15.37080643160202928953364359082, 16.2275129090407323765496506372, 16.90985546066379198442185114914, 17.92903305761770983412671841200, 19.43424247869860869679420636290, 19.94477984052163419538902144859, 21.51754781493030996380782300342, 22.78769658818457560609900606214, 23.36689382811076220627811889654, 24.088101947771816626460743451187, 25.783981732738519799971338979723, 26.36839129739414283769540376421, 27.26873929327566209437018231834, 27.940777542798864764853966586278

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