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Properties

Label 1-143-143.101-r1-0-0
Degree $1$
Conductor $143$
Sign $0.788 - 0.615i$
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  + (0.913 − 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (0.809 − 0.587i)5-s + (0.913 + 0.406i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (0.5 − 0.866i)10-s + 12-s + (0.309 − 0.951i)14-s + (0.978 + 0.207i)15-s + (−0.104 − 0.994i)16-s + (−0.913 − 0.406i)17-s + (0.309 + 0.951i)18-s + (−0.978 + 0.207i)19-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (0.809 − 0.587i)5-s + (0.913 + 0.406i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (0.5 − 0.866i)10-s + 12-s + (0.309 − 0.951i)14-s + (0.978 + 0.207i)15-s + (−0.104 − 0.994i)16-s + (−0.913 − 0.406i)17-s + (0.309 + 0.951i)18-s + (−0.978 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(15.3674\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 143,\ (1:\ ),\ 0.788 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.336230528 - 1.492848163i\)
\(L(\frac12)\) \(\approx\) \(4.336230528 - 1.492848163i\)
\(L(1)\) \(\approx\) \(2.529778794 - 0.5463565294i\)
\(L(1)\) \(\approx\) \(2.529778794 - 0.5463565294i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.913 - 0.406i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (0.669 - 0.743i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.978 + 0.207i)T \)
41 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.104 - 0.994i)T \)
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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

−28.50264810952901812905230971045, −26.66819277597428630781505913623, −25.873579401283228458386690377239, −24.99329787946153989975377825422, −24.4177941447137815750796300021, −23.37982588733325106192558967287, −22.13019195809136135030995814939, −21.35910070341634383466214881533, −20.45308459112978797090831748764, −19.128006967474582747748440703207, −17.97748014318991635455205898757, −17.26112059835293787823490290232, −15.47336384830386325989800730692, −14.72171847364798190292836749337, −13.93497960302655225004119889140, −13.006073822112396142615576275397, −12.013509926265238715360141469062, −10.75087121930814326702582532957, −8.9379486725567445498510617448, −7.978030302360314625149654647156, −6.63441052671388592652228953806, −5.95810265589855014210604326707, −4.358826038389367518560427246033, −2.65917566511035328465698685482, −2.03401016576059585734597480996, 1.46805262446751234410416815392, 2.705410406302003390290578831447, 4.2915518390933326779492698223, 4.85351861517194260685464601568, 6.3087366968863266637376733326, 7.99060646833749518379749925231, 9.37515431141705544440033988274, 10.34021878446954401377699324536, 11.27165229174094957321976333516, 12.84378909636037315849226075748, 13.77600874191181416667302318564, 14.36363600496174318246196392003, 15.563004749909938289468875278054, 16.57360451829186960580277269040, 17.79272509200581452392606177992, 19.57321632482455181392560989696, 20.2031624763427654702246264838, 21.20762570124627330001443036770, 21.55208736218596866489933562761, 22.857989324326627039670230304620, 24.029296364578027951332402169579, 24.90374209205100152975523934104, 25.78770417392445861437200086144, 27.10272389006780032097464581137, 28.0152434969248165810559616962

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