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Properties

Label 1-143-143.72-r0-0-0
Degree $1$
Conductor $143$
Sign $-0.780 - 0.625i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ -0.780 - 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4998433666 - 1.421862889i\)
\(L(\frac12)\) \(\approx\) \(0.4998433666 - 1.421862889i\)
\(L(1)\) \(\approx\) \(0.9416309015 - 1.061688251i\)
\(L(1)\) \(\approx\) \(0.9416309015 - 1.061688251i\)

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

−28.61097672793550727059960884332, −27.09149419793716324909212895058, −26.549143294299827047170058089080, −25.63160614632934449886980585448, −25.22470030866169076015626209337, −23.65116557350798937100176005179, −22.69522190952527339027525387785, −21.86599585607538934303539771821, −21.09782556792826461562721351216, −19.72559070239496295316113545720, −18.59739871877869709943257511329, −17.26918688808040244851377224425, −16.417678072661423705567228772868, −15.385271463001771703521607610777, −14.439271703739120199096279964211, −13.76862503390509058092182227849, −12.740644740238185308517213012014, −10.74469723147603204309386883504, −9.81847838946761517567162426208, −8.7556039946575371496847651197, −7.38576842821596636487711072313, −6.43752785620943020367842394297, −5.05792361753708404720100667714, −3.71138101575292635122462045831, −2.813943967121329192793854687062, 1.24376827076953763437460059555, 2.453759398865216167220101852740, 3.600462769339804563638211807831, 5.31946493517787291771404852226, 6.30923167275259413491318942879, 8.18504530360362247330281313240, 9.27188130640738509162710239656, 9.93423456160109070579884845738, 11.85890818756159507167197397410, 12.54083516675594386420880032656, 13.35355310437564322900131090127, 14.22413117003659381245154860039, 15.442521795805322915202806640890, 17.01098062357979419544619226139, 18.34040601683802568781326536792, 18.9975376361079677499692643056, 20.01171183752202074994191836191, 20.856430367148271395493971038, 21.68429578870680411700263845872, 22.98137037359047141429316168284, 23.89598424130771540899823945645, 25.04313987401164379733129445684, 25.58480352128695963910954954730, 27.158250842980336921001355677543, 28.25123593275680424459132443520

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