↑

Properties

Label 1-143-143.124-r1-0-0
Degree $1$
Conductor $143$
Sign $0.951 - 0.307i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.207 − 0.978i)2-s + (0.913 + 0.406i)3-s + (−0.913 + 0.406i)4-s + (−0.951 + 0.309i)5-s + (0.207 − 0.978i)6-s + (−0.406 − 0.913i)7-s + (0.587 + 0.809i)8-s + (0.669 + 0.743i)9-s + (0.5 + 0.866i)10-s − 12-s + (−0.809 + 0.587i)14-s + (−0.994 − 0.104i)15-s + (0.669 − 0.743i)16-s + (0.978 + 0.207i)17-s + (0.587 − 0.809i)18-s + (0.994 − 0.104i)19-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (0.913 + 0.406i)3-s + (−0.913 + 0.406i)4-s + (−0.951 + 0.309i)5-s + (0.207 − 0.978i)6-s + (−0.406 − 0.913i)7-s + (0.587 + 0.809i)8-s + (0.669 + 0.743i)9-s + (0.5 + 0.866i)10-s − 12-s + (−0.809 + 0.587i)14-s + (−0.994 − 0.104i)15-s + (0.669 − 0.743i)16-s + (0.978 + 0.207i)17-s + (0.587 − 0.809i)18-s + (0.994 − 0.104i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.951 - 0.307i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.951 - 0.307i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.649362481 - 0.2595292306i\)
\(L(\frac12)\) \(\approx\) \(1.649362481 - 0.2595292306i\)
\(L(1)\) \(\approx\) \(1.073279978 - 0.2555238425i\)
\(L(1)\) \(\approx\) \(1.073279978 - 0.2555238425i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−27.846044399699264462693372941324, −26.85271002929544399278951275359, −26.09002050501218439366196354081, −24.911767729994381072115588327198, −24.580959419956501309301758643164, −23.39008823925147669971113771248, −22.523198333047611745715056307683, −21.035101581437176502999234256030, −19.79358326496405323499207125439, −18.875141511792158441914542942778, −18.373239855002288520952901139694, −16.756937453787064490061974133409, −15.70526424454626595981460575950, −15.09945398806513888922844013538, −14.03516869073388966706398561226, −12.84712668795686696119171021347, −11.93742085280151121355363242257, −9.85408923299414576666093875146, −8.89714509549109790513165430829, −8.03711627343070444621177223168, −7.170723024339368731616513735499, −5.806693254466439156755318722997, −4.31467009070576720197347186783, −2.96756541157227149853802442026, −0.83466840640317401990102889792, 1.10294426981121947280924400513, 3.05257217774259245363703821831, 3.62074723483602033921305726903, 4.78860450989948074987698277104, 7.31546000613446643438836502678, 8.0696455019279931010544539104, 9.393150518356940428134325223945, 10.284287586135851535584026151820, 11.26119023834778753843577053483, 12.53695283545155926193344113047, 13.66088158910484725267639846929, 14.51252706867556619473076831599, 15.83024555492061012088325839991, 16.87347704355068238449848707865, 18.42695092799369438207978303216, 19.34476525500741494779043536325, 19.95162513447056906794751846365, 20.73936795147919513519040363358, 21.870802435935760429734809224565, 22.88907188604943899579406535988, 23.78032605090678972655919859117, 25.48760767411025279478855656143, 26.38883946738962645696844514601, 27.056831593266496025071250118307, 27.74272868550311786183048553954

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