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Properties

Label 1-143-143.80-r1-0-0
Degree $1$
Conductor $143$
Sign $-0.934 + 0.355i$
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  + (0.743 − 0.669i)2-s + (−0.104 − 0.994i)3-s + (0.104 − 0.994i)4-s + (0.951 − 0.309i)5-s + (−0.743 − 0.669i)6-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + (−0.978 + 0.207i)9-s + (0.5 − 0.866i)10-s − 12-s + (−0.809 + 0.587i)14-s + (−0.406 − 0.913i)15-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (−0.587 + 0.809i)18-s + (0.406 − 0.913i)19-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (−0.104 − 0.994i)3-s + (0.104 − 0.994i)4-s + (0.951 − 0.309i)5-s + (−0.743 − 0.669i)6-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + (−0.978 + 0.207i)9-s + (0.5 − 0.866i)10-s − 12-s + (−0.809 + 0.587i)14-s + (−0.406 − 0.913i)15-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (−0.587 + 0.809i)18-s + (0.406 − 0.913i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.934 + 0.355i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ -0.934 + 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3777985115 - 2.054442364i\)
\(L(\frac12)\) \(\approx\) \(-0.3777985115 - 2.054442364i\)
\(L(1)\) \(\approx\) \(0.8241908096 - 1.185047780i\)
\(L(1)\) \(\approx\) \(0.8241908096 - 1.185047780i\)

Euler product

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

−28.98834120165182291091418676019, −27.27474084137929124510304619559, −26.434061191679370408636301882470, −25.524355162769308266429044362686, −25.00277074294387978025938822954, −23.361253340532925049610532381711, −22.51143175591264010711964724017, −21.89908772334216630391779286344, −21.04971824562035025734284839323, −19.99544064751491222210361938960, −18.23691614052743853136604413709, −17.16686542496941180466870280093, −16.27379066451461887495476002416, −15.49755028547106610297173883529, −14.35212114882871431365165295935, −13.56173093157989971487203318486, −12.35409675568883170892667355513, −10.97885061830551115254244402377, −9.720188212259516288111030026237, −8.8853658367471134443841368713, −7.07082780480673337503765418774, −5.97667629057894260871580922483, −5.15817262890401255516515149899, −3.6813583348472371239276999856, −2.68929916143684782381629864017, 0.605746594560579812965862459950, 2.002667817051211101249021829929, 3.06519577172186740547319353798, 4.88345379628083230175078423970, 6.15754018131263341453521256493, 6.74957644629908770405597401661, 8.77176631837732916681849887350, 9.899844016986471766946599811062, 11.07892947975165732926508324896, 12.37952954437454947920195032847, 13.149385022004116154504155641266, 13.65027577647925023107372175571, 14.92359087250126169540808412562, 16.4359041136273278319964237595, 17.63846015699555963374346193529, 18.653655722951568910861539807782, 19.64700513403991522376627368963, 20.38525800325142316319655407070, 21.75820790677224076152973027958, 22.44723979089507824936550486699, 23.51584686941391392635587581854, 24.41899191487665767976649059705, 25.23628809640091264794780615572, 26.28022160620679614957596911495, 28.16186953616256473557078647301

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