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Properties

Label 1-143-143.24-r0-0-0
Degree $1$
Conductor $143$
Sign $0.996 - 0.0879i$
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  + (−0.406 − 0.913i)2-s + (0.669 + 0.743i)3-s + (−0.669 + 0.743i)4-s + (−0.587 − 0.809i)5-s + (0.406 − 0.913i)6-s + (0.743 + 0.669i)7-s + (0.951 + 0.309i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (0.207 − 0.978i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (0.951 − 0.309i)18-s + (0.207 + 0.978i)19-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (0.669 + 0.743i)3-s + (−0.669 + 0.743i)4-s + (−0.587 − 0.809i)5-s + (0.406 − 0.913i)6-s + (0.743 + 0.669i)7-s + (0.951 + 0.309i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (0.207 − 0.978i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (0.951 − 0.309i)18-s + (0.207 + 0.978i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.996 - 0.0879i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.996 - 0.0879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024486391 - 0.04515357924i\)
\(L(\frac12)\) \(\approx\) \(1.024486391 - 0.04515357924i\)
\(L(1)\) \(\approx\) \(0.9899347611 - 0.1173616002i\)
\(L(1)\) \(\approx\) \(0.9899347611 - 0.1173616002i\)

Euler product

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

−27.96040287409879664331389590442, −26.91910401020583829011912413366, −26.38785429362290860099067624895, −25.37916325509938785356810523729, −24.46236372219002446892237127322, −23.44989064741752388782488501896, −23.04457179131746977622944645286, −21.31090495846231741044672371883, −19.812500870013503530827485354696, −19.26683714244218078812885243135, −18.09499911530466063816349864112, −17.559913736496516813807784893821, −16.03936210982702205409228145903, −14.93607037902369743477045004812, −14.24807054819941502034767888100, −13.41948073961086459534908815422, −11.76059620565393865266523580351, −10.51572946002231287810899305307, −9.172185675681748882824344341922, −7.86323361250917618406052315801, −7.42723657725425946971646722526, −6.38018073332630179107161114124, −4.6533302679937199420048206978, −3.12446606764892732691039069807, −1.18571385808165310114602976391, 1.60029256159340869156690500833, 3.075653261794515663651224929326, 4.27492184330921470053049212576, 5.19982363368164182309374147332, 7.97295824533310471324682891186, 8.38934112729547753560260026689, 9.462630855956137952468559027267, 10.553037255455327325219220028002, 11.75294302143621148213110081643, 12.58138061364140718687474508954, 13.96280788333293444255911819309, 15.047006636239616388350053628906, 16.27320779348764711138587065014, 17.146325058740974024705202135924, 18.64902404731410340698906438070, 19.37660934244918014395386113680, 20.63708060189166339652138442147, 20.86983887357325422511446403676, 21.93664513112198906905736553245, 23.11190500613058845708280200107, 24.575885464851268276070760998988, 25.49206846471557128912235496376, 26.711449951856926624002555239213, 27.52219207867884544668775383287, 27.98522324339907638021497170240

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