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Properties

Label 2-143-11.4-c1-0-7
Degree $2$
Conductor $143$
Sign $0.761 - 0.647i$
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  + (1.91 + 1.38i)2-s + (0.562 − 1.73i)3-s + (1.10 + 3.40i)4-s + (−1.72 + 1.25i)5-s + (3.48 − 2.52i)6-s + (−0.333 − 1.02i)7-s + (−1.15 + 3.55i)8-s + (−0.256 − 0.186i)9-s − 5.04·10-s + (−2.78 − 1.80i)11-s + 6.52·12-s + (−0.809 − 0.587i)13-s + (0.786 − 2.42i)14-s + (1.20 + 3.69i)15-s + (−1.34 + 0.976i)16-s + (−0.957 + 0.695i)17-s + ⋯
L(s)  = 1  + (1.35 + 0.981i)2-s + (0.324 − 1.00i)3-s + (0.553 + 1.70i)4-s + (−0.772 + 0.561i)5-s + (1.42 − 1.03i)6-s + (−0.125 − 0.387i)7-s + (−0.408 + 1.25i)8-s + (−0.0856 − 0.0621i)9-s − 1.59·10-s + (−0.838 − 0.544i)11-s + 1.88·12-s + (−0.224 − 0.163i)13-s + (0.210 − 0.647i)14-s + (0.310 + 0.954i)15-s + (−0.336 + 0.244i)16-s + (−0.232 + 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.761 - 0.647i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.761 - 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89617 + 0.697237i\)
\(L(\frac12)\) \(\approx\) \(1.89617 + 0.697237i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.78 + 1.80i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-1.91 - 1.38i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.562 + 1.73i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.72 - 1.25i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.333 + 1.02i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (0.957 - 0.695i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.317 + 0.977i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5.23T + 23T^{2} \)
29 \( 1 + (-2.18 - 6.72i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.19 + 3.04i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.05 - 9.40i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.446 + 1.37i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 + (-2.39 + 7.35i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.19 + 4.49i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.87 - 11.9i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.15 + 3.01i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 4.78T + 67T^{2} \)
71 \( 1 + (12.3 - 8.94i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.55 + 4.79i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.522 - 0.379i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.368 + 0.267i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (9.53 + 6.93i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.36984760094476274611028665565, −12.69364333690673260225372694692, −11.70523954395999759706858877407, −10.43731767398878450771426773560, −8.289248372205744823866803538074, −7.48469370267476798890767915050, −6.90203936004280530155455720790, −5.65983266298934093523551414503, −4.17777172717750767430842463187, −2.89512613478228906454795176835, 2.54679852784340569278357651406, 4.01203211091956280819865089255, 4.55006920570342982610512135007, 5.77882200510457090677488380421, 7.78761540400312044076604894088, 9.241366195200617806022852137076, 10.21107927637750895379411144210, 11.14184890449783508262955654969, 12.30473204101759505650746953104, 12.64258259917585516218929489574

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