Properties

Label 1-143-143.71-r1-0-0
Degree $1$
Conductor $143$
Sign $0.289 + 0.957i$
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  + (0.406 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.289 + 0.957i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.289 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3359647830 + 0.2493537555i\)
\(L(\frac12)\) \(\approx\) \(0.3359647830 + 0.2493537555i\)
\(L(1)\) \(\approx\) \(0.8313497779 - 0.4185366282i\)
\(L(1)\) \(\approx\) \(0.8313497779 - 0.4185366282i\)

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

−27.51487492948757709154448209730, −26.59891314115334602641405504592, −26.015267426270713310524019920971, −24.84626312744346113813235058423, −24.07270323234716021214764235880, −22.88963074783699344740870694152, −22.135840388354856743290390418456, −20.84439919782662389607350365248, −20.081135428655274296299755020189, −18.988422940168580029311974876419, −17.23307967283064810201955939592, −16.443713902904415869464682359793, −15.68841907274340003564605782756, −14.86405920178391242151434012965, −13.468813745218353418744196167071, −12.997203654564943882710503318101, −11.334633291044100758277299840641, −9.68351434036711909260500153645, −8.82996702778886707921572947128, −7.8286832442745589106809699968, −6.63104406200697042787986297707, −4.89610092128799216788900457219, −4.22022197818199402520608938553, −3.03105678807952213694121927128, −0.1236485172522824010855736826, 1.868629996200915505125808405887, 3.00722730235881061889002160557, 3.84724404384703655844681814433, 5.85257269055518455671011941036, 6.96121122916614972454420913693, 8.44924407598243197724198930570, 9.490882167740018254226569648690, 10.79823001029214579436257031715, 11.97252592821130634475285207593, 12.73464198589996151428651817361, 13.79078186766941480454603722073, 14.82675190843296600343697300351, 15.590203054592571882323665039560, 17.71726412571428260230297389475, 18.741533218333800175219357902671, 19.22842854826620400328390910014, 20.03347239465630929750321355522, 21.26641183584738523881036197066, 22.36118235574422673251786023855, 23.13981469912555030620109967136, 24.11970910941481536897145466324, 25.30999981916691936809606894392, 26.353329279277101410991736199659, 27.29264918469541194686150545566, 28.575044686863720418969570436040

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