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Properties

Label 1-143-143.136-r1-0-0
Degree $1$
Conductor $143$
Sign $0.974 - 0.222i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.974 - 0.222i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.974 - 0.222i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9494538104 - 0.1069199974i\)
\(L(\frac12)\) \(\approx\) \(0.9494538104 - 0.1069199974i\)
\(L(1)\) \(\approx\) \(0.6994259500 - 0.03366951815i\)
\(L(1)\) \(\approx\) \(0.6994259500 - 0.03366951815i\)

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

−27.97907448235495429733294625079, −27.04552933336836053701917244999, −26.19311990438314830686692483189, −24.903494768470723575740167200680, −24.31921929134057324647735795278, −23.43328237714176270110248865679, −22.711885147909342286905950092442, −20.80885005591573397741262336882, −19.57168803607190918309581425096, −19.001538476934405636718107939280, −17.98114738840081569455130600582, −17.27013363857687123238513814481, −16.01324966003145163686217219496, −14.82053281734613445698251482891, −14.175688998232815845880850503306, −12.53942029300144618828140496597, −11.38653854548782702441096686999, −10.59231739929600905421173809862, −8.58479665068249920569321019720, −8.08884629203731400847979573730, −7.059708059298544593329393698711, −6.02620158541038354193774121034, −4.515892797001599531936121643229, −2.28583847080853559011446426626, −0.85155581571206230753765505732, 0.72922164042409037350218119439, 2.728921990974657565715613151358, 4.11224023140704964183307051350, 4.90742814587564224356069212830, 7.178540058358170666904209480453, 8.40709380181570785666552848701, 9.09411994322509552174444319922, 10.502890419930445287064890554118, 11.344013292076030947028947141885, 11.96728470445781748011148523203, 13.61332266614777908424534125400, 15.18583463114556834768813745756, 15.90766972324043590312468776244, 17.062196263810565037681247292979, 17.81649411439773850427125278437, 19.25357367020206684166999486693, 20.14248501257649201900888879127, 20.86441521436369654099916220255, 21.75454802271352506060359331195, 22.8556740158126988010199473645, 24.04208054996659770520601580444, 25.36627394241322130464319277503, 26.629677403796809116965156991192, 27.14174180168118613596512046467, 27.87554801927503805347190173238

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