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Properties

Label 2-143-11.5-c1-0-3
Degree $2$
Conductor $143$
Sign $0.636 + 0.771i$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.620 − 1.91i)2-s + (1.77 + 1.29i)3-s + (−1.64 + 1.19i)4-s + (−0.772 + 2.37i)5-s + (1.36 − 4.19i)6-s + (3.01 − 2.18i)7-s + (0.0562 + 0.0408i)8-s + (0.561 + 1.72i)9-s + 5.01·10-s + (3.29 − 0.355i)11-s − 4.46·12-s + (−0.309 − 0.951i)13-s + (−6.05 − 4.39i)14-s + (−4.43 + 3.22i)15-s + (−1.21 + 3.73i)16-s + (−1.47 + 4.55i)17-s + ⋯
L(s)  = 1  + (−0.438 − 1.35i)2-s + (1.02 + 0.744i)3-s + (−0.823 + 0.597i)4-s + (−0.345 + 1.06i)5-s + (0.556 − 1.71i)6-s + (1.13 − 0.827i)7-s + (0.0198 + 0.0144i)8-s + (0.187 + 0.576i)9-s + 1.58·10-s + (0.994 − 0.107i)11-s − 1.28·12-s + (−0.0857 − 0.263i)13-s + (−1.61 − 1.17i)14-s + (−1.14 + 0.832i)15-s + (−0.303 + 0.934i)16-s + (−0.358 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.636 + 0.771i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.636 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09079 - 0.514031i\)
\(L(\frac12)\) \(\approx\) \(1.09079 - 0.514031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-3.29 + 0.355i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (0.620 + 1.91i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.77 - 1.29i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.772 - 2.37i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-3.01 + 2.18i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (1.47 - 4.55i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.41 + 3.93i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.70T + 23T^{2} \)
29 \( 1 + (7.25 - 5.27i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.439 + 1.35i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.63 - 1.18i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.08 + 2.24i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 + (1.69 + 1.22i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.660 + 2.03i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.98 + 4.35i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.30 + 10.1i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 1.03T + 67T^{2} \)
71 \( 1 + (1.56 - 4.83i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.73 - 3.43i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.55 - 10.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.539 + 1.66i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 4.32T + 89T^{2} \)
97 \( 1 + (-4.24 - 13.0i)T + (-78.4 + 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.84143196044581392915137014350, −11.40988585745393218562315591796, −10.89433174460674177074046801240, −10.17286511132301970181825502228, −9.022146934055874714954430654303, −8.190997143644336740348661515649, −6.74103859533715053498068709142, −4.16252908418355769859736468037, −3.54048392853349386679808328386, −2.04176052654027028024596142974, 1.98959378250340273813390609950, 4.49781190436570625956846829524, 5.81983247248612833918500090556, 7.20391282284443324120725070383, 8.124949689987350883704230542668, 8.648208172879364353138473397418, 9.300991468694288067509207512171, 11.59793730720916688194032157165, 12.32421028274600390586118864736, 13.60930680851330511504578081511

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