Properties

Label 1-143-143.139-r1-0-0
Degree $1$
Conductor $143$
Sign $0.286 - 0.958i$
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  + (−0.669 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.286 - 0.958i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.286 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2351491964 - 0.1751195589i\)
\(L(\frac12)\) \(\approx\) \(0.2351491964 - 0.1751195589i\)
\(L(1)\) \(\approx\) \(0.5497644900 + 0.2377634347i\)
\(L(1)\) \(\approx\) \(0.5497644900 + 0.2377634347i\)

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

−28.444958266672143008154838348882, −27.315931535157627709039805295, −26.069980795219052910758956343311, −25.83173212691290878312383067428, −24.32542302579641337980583001201, −23.20601187776601013380887980550, −22.2864729601519956435571354790, −21.193632220754555544898145252339, −19.83297866386767583255369338702, −19.34567421061756301828956819047, −18.16497512215148937569172514626, −17.56881277045645906258183296581, −16.67864628426406673761617565191, −14.79753269757587280338263101187, −13.53417719244965063427818543557, −12.934837032804829138962636652575, −11.37250088070410500876644982490, −10.864662593643337861530953630148, −9.620806368023069086946161194048, −8.11217828593103690677722798999, −7.2515874480587259140762524360, −6.255130630834008821011752821512, −4.01805905920855581123967526527, −2.61649406864887926171472323656, −1.48720968991940164613092005925, 0.14034058436026666096965713629, 2.16945888509184579663656971042, 4.4324609315031160747405670707, 5.35072751735116875393554378482, 6.2915987360736521298887271254, 8.24474545436454795610389291559, 8.95001386847106758571982158690, 9.7437190537568520486692501940, 10.96492613585916652021289438550, 12.28143523143726059854709284779, 13.86987200287936888649210852105, 15.01759064501547451003897842003, 15.82297498040906241586042867241, 16.63309942877337027949119459574, 17.512915482445095830976693840255, 18.598043752357818186793650000416, 19.93953354406968618664195538165, 20.80690375640557985613697647585, 21.887463909236983631910787952390, 22.96062536156138502864000242562, 24.2264833206338087752709675128, 25.039203276019161659958818764962, 25.84483378597630843725276957344, 26.99093341553354889504498179698, 27.81014819798665997096105954477

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