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Properties

Label 1-143-143.60-r1-0-0
Degree $1$
Conductor $143$
Sign $-0.999 + 0.0255i$
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  + (−0.587 − 0.809i)2-s + (0.309 + 0.951i)3-s + (−0.309 + 0.951i)4-s + (0.587 − 0.809i)5-s + (0.587 − 0.809i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s − 10-s − 12-s + (0.309 + 0.951i)14-s + (0.951 + 0.309i)15-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (0.951 + 0.309i)18-s + (−0.951 + 0.309i)19-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (0.309 + 0.951i)3-s + (−0.309 + 0.951i)4-s + (0.587 − 0.809i)5-s + (0.587 − 0.809i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s − 10-s − 12-s + (0.309 + 0.951i)14-s + (0.951 + 0.309i)15-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (0.951 + 0.309i)18-s + (−0.951 + 0.309i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.999 + 0.0255i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ -0.999 + 0.0255i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002592314604 - 0.2025004032i\)
\(L(\frac12)\) \(\approx\) \(0.002592314604 - 0.2025004032i\)
\(L(1)\) \(\approx\) \(0.6384814238 - 0.1326288473i\)
\(L(1)\) \(\approx\) \(0.6384814238 - 0.1326288473i\)

Euler product

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

−28.69608775229063271863201916870, −27.38865975432036754396582920492, −26.05799487720204038769427012061, −25.66370022356872402839232328253, −24.9740367196985190817589625024, −23.72636610162079011522326318561, −22.940856528349083761590418277517, −21.88966748952428017241606566099, −20.13183907139294153114292579754, −19.111220204501874223713809971882, −18.52172782119043399952217956703, −17.67268478859103384280220741695, −16.57565401151324199793456711361, −15.30321475899451003772860985072, −14.27616258252738335301627140582, −13.53332860426573004554225690155, −12.2636892311772781928826173610, −10.61626603177786032005944674451, −9.560414247964102231874619569757, −8.51948893931291505498433060500, −7.16976752649061888449941576960, −6.52243741120228895748909915174, −5.54987468709065626379822404599, −3.13952932060686623472769747614, −1.75898214191725722169972326570, 0.08591888337041083211786584435, 1.97485924954370753870421950961, 3.43944337150156169052830753673, 4.420049234854787086909295550776, 5.98170709554204472085795966904, 7.982896825778878034369999302575, 8.9685654413794602715427557073, 9.909629737664442247245032909130, 10.43941103877631866263132821189, 12.02148423733415860816495131764, 13.051080456859492972413415362318, 14.02046445289105428206648478572, 15.674701345043788809732605327, 16.75564790418884361907475918279, 17.13297647141338335129410015695, 18.788699454141879463904256635153, 19.79721913941013725312250006699, 20.52105792665241297262723969658, 21.415856446344080847507308736711, 22.13312958807132869931050299116, 23.34003706052628793797960335300, 25.15968087147052310985268446004, 25.8092427810221963410264058912, 26.60862621993179699631364513531, 27.84475752118021693445670590875

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