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Properties

Label 1-143-143.36-r0-0-0
Degree $1$
Conductor $143$
Sign $-0.939 + 0.343i$
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  + (−0.669 − 0.743i)2-s + (−0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (−0.309 − 0.951i)5-s + (−0.669 + 0.743i)6-s + (0.104 − 0.994i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + 12-s + (−0.809 + 0.587i)14-s + (−0.913 + 0.406i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (0.809 + 0.587i)18-s + (−0.913 − 0.406i)19-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (−0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (−0.309 − 0.951i)5-s + (−0.669 + 0.743i)6-s + (0.104 − 0.994i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.5 + 0.866i)10-s + 12-s + (−0.809 + 0.587i)14-s + (−0.913 + 0.406i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (0.809 + 0.587i)18-s + (−0.913 − 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.939 + 0.343i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ -0.939 + 0.343i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09683081115 - 0.5464526503i\)
\(L(\frac12)\) \(\approx\) \(-0.09683081115 - 0.5464526503i\)
\(L(1)\) \(\approx\) \(0.3628733444 - 0.5217913923i\)
\(L(1)\) \(\approx\) \(0.3628733444 - 0.5217913923i\)

Euler product

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

−28.29939021502992911973075740677, −27.737306280147766625199698706562, −26.86460031047529703507787561312, −25.94209968121719753988717258616, −25.3527236461189169763222826127, −23.92742785200269559989599731654, −22.88823909471717733125705247303, −22.08154771432641829556511293399, −20.99433282306696240927080705004, −19.55026168459903124493651587240, −18.73045805877095628372916196344, −17.733057043502116902744279970456, −16.6149262921612750928058723361, −15.61994700007398303451490781654, −14.89429047078900289087667689537, −14.25511007875396643975489346837, −12.100081406571101063507357831048, −10.81596241389389667337506151654, −10.15792704875107301175907622057, −8.89981372121065813863254417107, −8.00327733203918328145408149107, −6.41543531905400317605914718024, −5.58295428770416773887588810607, −4.09341035327601984233778315017, −2.43794463942360003823512095195, 0.62082670352866547380227195675, 1.78740456671484037388754789301, 3.49083588869431635342569907050, 4.93632879235822814744506094639, 6.87568225609765728650972130388, 7.85271150710520382556043778597, 8.69372518448888776349202984892, 10.05932301889187221606918968451, 11.359591207383959002598178019299, 12.182956934908107709005334405152, 13.13997567109348648040179676366, 13.9525519542968356962860588110, 16.04129936402151959699348061099, 17.05055879319865124425207061201, 17.6278228419293182671922322568, 18.90163261665618361787134985302, 19.77517212892267610161019439605, 20.37046797238066733112974823664, 21.48314116273145481571324737217, 23.06158540355281006713994024001, 23.70149753129204076280672137374, 24.939010297876052332209186856665, 25.75698778752232564158241887360, 27.035698192663746073137971766481, 27.87671045091180772854432308874

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