↑

Properties

Label 2-143-13.3-c3-0-0
Degree $2$
Conductor $143$
Sign $0.249 - 0.968i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.706 − 1.22i)2-s + (−2.88 − 4.99i)3-s + (3.00 − 5.19i)4-s − 4.58·5-s + (−4.07 + 7.05i)6-s + (−18.2 + 31.5i)7-s − 19.7·8-s + (−3.11 + 5.40i)9-s + (3.24 + 5.61i)10-s + (−5.5 − 9.52i)11-s − 34.5·12-s + (23.5 + 40.5i)13-s + 51.4·14-s + (13.2 + 22.9i)15-s + (−10.0 − 17.3i)16-s + (33.3 − 57.7i)17-s + ⋯
L(s)  = 1  + (−0.249 − 0.432i)2-s + (−0.554 − 0.960i)3-s + (0.375 − 0.649i)4-s − 0.410·5-s + (−0.277 + 0.480i)6-s + (−0.983 + 1.70i)7-s − 0.874·8-s + (−0.115 + 0.200i)9-s + (0.102 + 0.177i)10-s + (−0.150 − 0.261i)11-s − 0.832·12-s + (0.502 + 0.864i)13-s + 0.983·14-s + (0.227 + 0.394i)15-s + (−0.156 − 0.270i)16-s + (0.475 − 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.249 - 0.968i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ 0.249 - 0.968i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.166571 + 0.129076i\)
\(L(\frac12)\) \(\approx\) \(0.166571 + 0.129076i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (5.5 + 9.52i)T \)
13 \( 1 + (-23.5 - 40.5i)T \)
good2 \( 1 + (0.706 + 1.22i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (2.88 + 4.99i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 4.58T + 125T^{2} \)
7 \( 1 + (18.2 - 31.5i)T + (-171.5 - 297. i)T^{2} \)
17 \( 1 + (-33.3 + 57.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (9.57 - 16.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-73.9 - 128. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-8.97 - 15.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 272.T + 2.97e4T^{2} \)
37 \( 1 + (-122. - 212. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-22.3 - 38.6i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (15.5 - 26.8i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 285.T + 1.03e5T^{2} \)
53 \( 1 + 436.T + 1.48e5T^{2} \)
59 \( 1 + (434. - 752. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (93.2 - 161. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (344. + 596. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-59.5 + 103. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 786.T + 3.89e5T^{2} \)
79 \( 1 - 254.T + 4.93e5T^{2} \)
83 \( 1 + 768.T + 5.71e5T^{2} \)
89 \( 1 + (-63.1 - 109. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-679. + 1.17e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.46440833224281357889303074906, −11.81869497119739943349734136748, −11.25645887908454630912809718051, −9.623804173188342379273893242005, −9.002589558096522381735121342126, −7.30659043834456470930038362810, −6.19179909646904092257693839188, −5.60847753769723394230232079014, −3.07845270573184257947814333978, −1.62668200379196343223253580067, 0.11724396482060438792454470647, 3.43399780063034892476050303903, 4.20962922007609801378916690756, 5.96094601872479086754879707450, 7.15823526532651106367098505416, 7.960128036662835803945633913774, 9.469735045089363138218710009173, 10.58702908328086977283236471368, 10.96786637566770454085362257789, 12.55796837413399248060310640295

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