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Properties

Label 2-143-11.9-c1-0-3
Degree $2$
Conductor $143$
Sign $0.987 - 0.155i$
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  + (0.481 − 1.48i)2-s + (−1.52 + 1.11i)3-s + (−0.347 − 0.252i)4-s + (1.25 + 3.85i)5-s + (0.910 + 2.80i)6-s + (0.257 + 0.186i)7-s + (1.98 − 1.43i)8-s + (0.177 − 0.547i)9-s + 6.31·10-s + (−2.23 − 2.45i)11-s + 0.811·12-s + (0.309 − 0.951i)13-s + (0.400 − 0.291i)14-s + (−6.20 − 4.50i)15-s + (−1.44 − 4.44i)16-s + (0.908 + 2.79i)17-s + ⋯
L(s)  = 1  + (0.340 − 1.04i)2-s + (−0.883 + 0.641i)3-s + (−0.173 − 0.126i)4-s + (0.560 + 1.72i)5-s + (0.371 + 1.14i)6-s + (0.0971 + 0.0705i)7-s + (0.700 − 0.508i)8-s + (0.0592 − 0.182i)9-s + 1.99·10-s + (−0.673 − 0.739i)11-s + 0.234·12-s + (0.0857 − 0.263i)13-s + (0.107 − 0.0777i)14-s + (−1.60 − 1.16i)15-s + (−0.361 − 1.11i)16-s + (0.220 + 0.678i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.987 - 0.155i$
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: \((2,\ 143,\ (\ :1/2),\ 0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15761 + 0.0903171i\)
\(L(\frac12)\) \(\approx\) \(1.15761 + 0.0903171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.23 + 2.45i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (-0.481 + 1.48i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.52 - 1.11i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-1.25 - 3.85i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.257 - 0.186i)T + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (-0.908 - 2.79i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.91 + 1.38i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 5.05T + 23T^{2} \)
29 \( 1 + (1.24 + 0.905i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.76 + 5.42i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.17 + 3.03i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.15 + 3.74i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.12T + 43T^{2} \)
47 \( 1 + (6.86 - 4.98i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.91 - 5.90i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.82 - 7.14i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.77 + 11.6i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.94T + 67T^{2} \)
71 \( 1 + (-0.0667 - 0.205i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-12.0 - 8.76i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.67 + 11.3i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.966 - 2.97i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 2.04T + 89T^{2} \)
97 \( 1 + (4.08 - 12.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.08838188423864527181500709156, −11.66910030871073754738761932742, −10.93470823339801145024991813561, −10.62969944664905765796212967202, −9.747850882147961327137786238005, −7.67531296228291579544094866449, −6.39131153739269293225591681751, −5.28461121930202737387106133920, −3.58692088353714537960122336771, −2.50594628824345088645740727804, 1.43275241151041111345201105618, 4.92007712170399224399151783985, 5.26978191003982420120776289493, 6.45671553401920322173138755256, 7.49158807735924856558567462254, 8.655550663277696548396377025436, 9.880115176631121858523143766556, 11.36632108234725073720477426402, 12.39643232726005680531124141339, 13.06484656728017309167897821660

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