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Properties

Label 1-143-143.123-r0-0-0
Degree $1$
Conductor $143$
Sign $-0.350 + 0.936i$
Analytic cond. $0.664089$
Root an. cond. $0.664089$
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  + (−0.994 − 0.104i)2-s + (−0.978 + 0.207i)3-s + (0.978 + 0.207i)4-s + (0.587 + 0.809i)5-s + (0.994 − 0.104i)6-s + (−0.207 + 0.978i)7-s + (−0.951 − 0.309i)8-s + (0.913 − 0.406i)9-s + (−0.5 − 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (−0.743 − 0.669i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (−0.951 + 0.309i)18-s + (−0.743 + 0.669i)19-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−0.978 + 0.207i)3-s + (0.978 + 0.207i)4-s + (0.587 + 0.809i)5-s + (0.994 − 0.104i)6-s + (−0.207 + 0.978i)7-s + (−0.951 − 0.309i)8-s + (0.913 − 0.406i)9-s + (−0.5 − 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (−0.743 − 0.669i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (−0.951 + 0.309i)18-s + (−0.743 + 0.669i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.350 + 0.936i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ -0.350 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2640641247 + 0.3808448697i\)
\(L(\frac12)\) \(\approx\) \(0.2640641247 + 0.3808448697i\)
\(L(1)\) \(\approx\) \(0.4936429505 + 0.2057225502i\)
\(L(1)\) \(\approx\) \(0.4936429505 + 0.2057225502i\)

Euler product

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

−28.29029327556704994389865604791, −27.12619171749906942672825158424, −26.155685724959877078435743986483, −25.04490196948200669463412130707, −24.072295761290347841094167412874, −23.44587484016192981335684086374, −21.92460103382321514018832103914, −20.88008285956279725813097671563, −19.89492798361314811016003524485, −18.80472108406420282530941340743, −17.633838813122576218685140081783, −16.93467208686606131588889719894, −16.499713245590798026678680652518, −15.11575706297391829407766808010, −13.340621563571111238090638874430, −12.49080622131528843703024123175, −11.09667951581798649989530146350, −10.34691660508661929527041832877, −9.270753361875399984129813597841, −7.93007928077129349567235908232, −6.71679231887185535354928540666, −5.82522925694319288859504907206, −4.348635140078628259289634331770, −1.94735210102031649827204296785, −0.62819300417988897244714403058, 1.77392717605992739707273594329, 3.18743346836701595715007867559, 5.41010098594901555028791203265, 6.34457457090934379913904578627, 7.312720785784006408620331186290, 9.04244598384972077579311291848, 9.89497584024254319599236080488, 10.914686460247841827011068382571, 11.7369301154039945108076649706, 12.86180832316233486979548049414, 14.7431548412138919633988186346, 15.691055396098502979428474582953, 16.67625902730050019514436614715, 17.73779250144238112611830884339, 18.39936208190134756366568890363, 19.13478531691788850684061541519, 20.79341951022225226176689624526, 21.6716963950598313887876293614, 22.41526365064727249694292080652, 23.7017371900226410266262388795, 25.05829906972316083507788808774, 25.615574774802974890453805516023, 26.9494294162693269479187288274, 27.54951193969001882177209983778, 28.65583261712891545074074687209

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