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Properties

Label 2-143-11.9-c1-0-5
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: $\chi_{143} (53, \cdot )$
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

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

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