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Properties

Label 1-143-143.38-r0-0-0
Degree $1$
Conductor $143$
Sign $0.964 + 0.265i$
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  + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + 10-s + 12-s + (0.309 + 0.951i)14-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)18-s + (−0.309 − 0.951i)19-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + 10-s + 12-s + (0.309 + 0.951i)14-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)18-s + (−0.309 − 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(0.664089\)
Root analytic conductor: \(0.664089\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (0:\ ),\ 0.964 + 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.886222678 + 0.2546613688i\)
\(L(\frac12)\) \(\approx\) \(1.886222678 + 0.2546613688i\)
\(L(1)\) \(\approx\) \(1.740397797 + 0.08475182329i\)
\(L(1)\) \(\approx\) \(1.740397797 + 0.08475182329i\)

Euler product

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

−28.77689906556051991042912394175, −26.86394583188589995033715201734, −25.8680079397974037753439840523, −25.149161596629434905546759366335, −24.31968475728305822734074417547, −23.49594348140967702004220293087, −22.62508239731322414107668540562, −21.26257220450938616910408089382, −20.441681101143701227511984724205, −19.44206990944816401795606610524, −17.81391162826051428011592394415, −17.17553295201078738476341406869, −16.21029779299586599570857229003, −14.64091058631613377894482750044, −13.83931300602526806918183004162, −13.01827271199565210377125477513, −12.40724852679420271993591586352, −10.79370453294874704819507544046, −9.07020779707982328636487843990, −8.00165927775229704384080728999, −6.81028382342719733528711473904, −6.03764479234011000039813530056, −4.603868496788767810135624728351, −3.13587534350305015869331114231, −1.61470811451657133080588744696, 2.35893887996558478099649173383, 2.97867548231672152663672141224, 4.557639638626947876015963048870, 5.58425473385700245338895090393, 6.6653256225338118016092549232, 8.94042141425790423504054578283, 9.70481427945259393806950526712, 10.76442034782587981517762021830, 11.675142305110714200529580662770, 13.19838285861419905611523847788, 13.9796666714458545311860303451, 15.203291644527540898314708397381, 15.56406987532350740389740381954, 17.229466198561090311988604620076, 18.61766597263153108968646225930, 19.5263136506946573413251884844, 20.768270068660098383698944656202, 21.45914909563681067343503407608, 22.25004071505372329824537034453, 22.81095985878247945016113593174, 24.51669145561170883530686484285, 25.33725907786765514275419549909, 26.329732580839071912328767641188, 27.55359342875669049627740574432, 28.562241347997928214835600274729

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