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Properties

Label 1-143-143.93-r1-0-0
Degree $1$
Conductor $143$
Sign $-0.240 + 0.970i$
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  + (−0.994 + 0.104i)2-s + (−0.978 − 0.207i)3-s + (0.978 − 0.207i)4-s + (−0.587 + 0.809i)5-s + (0.994 + 0.104i)6-s + (−0.207 − 0.978i)7-s + (−0.951 + 0.309i)8-s + (0.913 + 0.406i)9-s + (0.5 − 0.866i)10-s − 12-s + (0.309 + 0.951i)14-s + (0.743 − 0.669i)15-s + (0.913 − 0.406i)16-s + (0.104 − 0.994i)17-s + (−0.951 − 0.309i)18-s + (−0.743 − 0.669i)19-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (−0.978 − 0.207i)3-s + (0.978 − 0.207i)4-s + (−0.587 + 0.809i)5-s + (0.994 + 0.104i)6-s + (−0.207 − 0.978i)7-s + (−0.951 + 0.309i)8-s + (0.913 + 0.406i)9-s + (0.5 − 0.866i)10-s − 12-s + (0.309 + 0.951i)14-s + (0.743 − 0.669i)15-s + (0.913 − 0.406i)16-s + (0.104 − 0.994i)17-s + (−0.951 − 0.309i)18-s + (−0.743 − 0.669i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.240 + 0.970i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ -0.240 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1847725716 + 0.2360753186i\)
\(L(\frac12)\) \(\approx\) \(0.1847725716 + 0.2360753186i\)
\(L(1)\) \(\approx\) \(0.4196514215 + 0.02149661258i\)
\(L(1)\) \(\approx\) \(0.4196514215 + 0.02149661258i\)

Euler product

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

−27.79209344755342839384547502970, −27.26202547087101972009986448371, −25.91604504407481648397346573474, −24.79293399741106362490903538542, −24.0057809350817753918852423798, −22.90730268095818376946770419445, −21.500579487684371694582294135872, −20.921534552367790985767235435981, −19.41149319539505195145005662057, −18.81517243540489106849014084149, −17.4855477517136894017691393035, −16.86081106634031403916747466912, −15.75131682190366513850712482823, −15.248614884290447352031963400334, −12.73522413788353408955118933840, −12.11604914275673433151337504312, −11.19699902168793107218870812697, −10.00140634482994180236867599977, −8.92215895522804817549823063517, −7.90614899534129844425667303014, −6.419457563070023542568335917422, −5.40942231877068626283383736226, −3.78660531816340635894292166214, −1.762154362237056654708304394157, −0.23319809803882715851285150311, 0.98408047879850427757185844806, 2.92487879809748750559088993298, 4.70767284129839924668492494576, 6.60452300158633920342847347160, 6.92607547297033622198549855803, 8.13984616085317312629101945697, 9.86348317331947745276045015915, 10.754729273592940003757513266481, 11.37743052446406896019474631100, 12.59384142738515186635599182390, 14.21622082137718129236398713235, 15.58529810271846663638741670115, 16.3833257974911225130022611135, 17.350112061936121052840688312173, 18.225687680254267695881987135347, 19.11593938575850369170605356011, 20.00382070748171461961677909410, 21.340923330507721721607508936388, 22.707549490680497236846803227564, 23.39551765047414673664337569340, 24.33092940029342645187206326862, 25.6010176305459499262429413876, 26.71482394893286099679183746201, 27.21987771995118626498010596687, 28.22305105727009511348890188159

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