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Properties

Label 2-143-13.3-c1-0-5
Degree $2$
Conductor $143$
Sign $0.997 - 0.0645i$
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  + (0.249 + 0.432i)2-s + (0.120 + 0.208i)3-s + (0.875 − 1.51i)4-s + 0.581·5-s + (−0.0600 + 0.103i)6-s + (−0.705 + 1.22i)7-s + 1.87·8-s + (1.47 − 2.54i)9-s + (0.145 + 0.251i)10-s + (0.5 + 0.866i)11-s + 0.421·12-s + (−2.39 + 2.69i)13-s − 0.703·14-s + (0.0699 + 0.121i)15-s + (−1.28 − 2.22i)16-s + (−0.225 + 0.391i)17-s + ⋯
L(s)  = 1  + (0.176 + 0.305i)2-s + (0.0694 + 0.120i)3-s + (0.437 − 0.758i)4-s + 0.260·5-s + (−0.0245 + 0.0424i)6-s + (−0.266 + 0.461i)7-s + 0.661·8-s + (0.490 − 0.849i)9-s + (0.0458 + 0.0794i)10-s + (0.150 + 0.261i)11-s + 0.121·12-s + (−0.664 + 0.747i)13-s − 0.188·14-s + (0.0180 + 0.0312i)15-s + (−0.321 − 0.556i)16-s + (−0.0547 + 0.0948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.997 - 0.0645i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.997 - 0.0645i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36234 + 0.0440083i\)
\(L(\frac12)\) \(\approx\) \(1.36234 + 0.0440083i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (2.39 - 2.69i)T \)
good2 \( 1 + (-0.249 - 0.432i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.120 - 0.208i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.581T + 5T^{2} \)
7 \( 1 + (0.705 - 1.22i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (0.225 - 0.391i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.59 - 2.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.01 - 3.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.10 + 8.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.10T + 31T^{2} \)
37 \( 1 + (2.34 + 4.05i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.43 - 7.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.20 - 3.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.77T + 47T^{2} \)
53 \( 1 + 0.0779T + 53T^{2} \)
59 \( 1 + (-3.85 + 6.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.16 + 3.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.61 - 9.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.992 + 1.71i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.84T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 4.03T + 83T^{2} \)
89 \( 1 + (-5.11 - 8.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.25 + 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.18975684515644100424189133171, −12.11150816775497705462047827056, −11.14338326233771913617835878550, −9.723313803316278699543046037404, −9.431566142680580184716931215715, −7.56324565712822781504747564402, −6.49732579470110048669667273040, −5.59090468971908884798489462201, −4.08024199005731904581238150401, −1.98750232469379901628797334512, 2.23828278123206340757799759845, 3.70249348520767051997521891163, 5.16780016978284638267533579429, 6.93370609716848368547878962160, 7.61398475433966870782643818730, 8.894828783079079806542117658202, 10.38081729346778328969318391779, 10.97015203504874996024285451435, 12.30773128964235975280305386524, 13.03514141504376821591983355535

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