Properties

Label 2-143-13.12-c3-0-22
Degree $2$
Conductor $143$
Sign $0.997 + 0.0632i$
Analytic cond. $8.43727$
Root an. cond. $2.90469$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.51i·2-s + 3.00·3-s − 22.4·4-s − 18.8i·5-s + 16.5i·6-s − 0.324i·7-s − 79.5i·8-s − 17.9·9-s + 104.·10-s + 11i·11-s − 67.3·12-s + (2.96 − 46.7i)13-s + 1.79·14-s − 56.7i·15-s + 259.·16-s + 85.7·17-s + ⋯
L(s)  = 1  + 1.94i·2-s + 0.578·3-s − 2.80·4-s − 1.68i·5-s + 1.12i·6-s − 0.0175i·7-s − 3.51i·8-s − 0.665·9-s + 3.29·10-s + 0.301i·11-s − 1.62·12-s + (0.0632 − 0.997i)13-s + 0.0342·14-s − 0.976i·15-s + 4.05·16-s + 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0632i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.997 + 0.0632i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ 0.997 + 0.0632i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.08563 - 0.0343483i\)
\(L(\frac12)\) \(\approx\) \(1.08563 - 0.0343483i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - 11iT \)
13 \( 1 + (-2.96 + 46.7i)T \)
good2 \( 1 - 5.51iT - 8T^{2} \)
3 \( 1 - 3.00T + 27T^{2} \)
5 \( 1 + 18.8iT - 125T^{2} \)
7 \( 1 + 0.324iT - 343T^{2} \)
17 \( 1 - 85.7T + 4.91e3T^{2} \)
19 \( 1 + 62.7iT - 6.85e3T^{2} \)
23 \( 1 + 108.T + 1.21e4T^{2} \)
29 \( 1 + 149.T + 2.43e4T^{2} \)
31 \( 1 + 265. iT - 2.97e4T^{2} \)
37 \( 1 - 120. iT - 5.06e4T^{2} \)
41 \( 1 + 49.7iT - 6.89e4T^{2} \)
43 \( 1 - 38.8T + 7.95e4T^{2} \)
47 \( 1 + 94.6iT - 1.03e5T^{2} \)
53 \( 1 - 237.T + 1.48e5T^{2} \)
59 \( 1 + 282. iT - 2.05e5T^{2} \)
61 \( 1 + 359.T + 2.26e5T^{2} \)
67 \( 1 - 325. iT - 3.00e5T^{2} \)
71 \( 1 + 658. iT - 3.57e5T^{2} \)
73 \( 1 - 1.02e3iT - 3.89e5T^{2} \)
79 \( 1 - 781.T + 4.93e5T^{2} \)
83 \( 1 + 822. iT - 5.71e5T^{2} \)
89 \( 1 - 976. iT - 7.04e5T^{2} \)
97 \( 1 - 1.76e3iT - 9.12e5T^{2} \)
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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

−13.14992822969097563166513149933, −12.18731819414942531054397941299, −9.787964964530539320327284609095, −9.053738909939842620832725195009, −8.168868568094892773385941695299, −7.66762427864037510660113499757, −5.84312119205885434071294117257, −5.22845456593470033429614796821, −3.94620617557181293254103198123, −0.51028911261946340220709631819, 1.97461385250650625841094943174, 3.09577601753574370930332027338, 3.79914191494811583142713307842, 5.79766335878283437911478681969, 7.71263567176796801940011629536, 8.912885999401738223924181954915, 9.942806521757023543990266186433, 10.69777884166504190405402688975, 11.52948735223525285577028892441, 12.27520362246575764872547646871

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