Properties

Degree $2$
Conductor $1143$
Sign $-0.577 - 0.816i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.51i·2-s − 2.34·4-s + 1.17i·5-s + 3.92·7-s − 4.15i·8-s + 2.96·10-s + 0.116i·11-s − 19.8·13-s − 9.87i·14-s − 19.8·16-s − 4.09i·17-s + 0.981·19-s − 2.76i·20-s + 0.292·22-s − 3.96i·23-s + ⋯
L(s)  = 1  − 1.25i·2-s − 0.587·4-s + 0.235i·5-s + 0.560·7-s − 0.519i·8-s + 0.296·10-s + 0.0105i·11-s − 1.52·13-s − 0.705i·14-s − 1.24·16-s − 0.240i·17-s + 0.0516·19-s − 0.138i·20-s + 0.0133·22-s − 0.172i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1143\)    =    \(3^{2} \cdot 127\)
Sign: $-0.577 - 0.816i$
Motivic weight: \(2\)
Character: $\chi_{1143} (890, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1143,\ (\ :1),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7351467256\)
\(L(\frac12)\) \(\approx\) \(0.7351467256\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
127 \( 1 + 11.2T \)
good2 \( 1 + 2.51iT - 4T^{2} \)
5 \( 1 - 1.17iT - 25T^{2} \)
7 \( 1 - 3.92T + 49T^{2} \)
11 \( 1 - 0.116iT - 121T^{2} \)
13 \( 1 + 19.8T + 169T^{2} \)
17 \( 1 + 4.09iT - 289T^{2} \)
19 \( 1 - 0.981T + 361T^{2} \)
23 \( 1 + 3.96iT - 529T^{2} \)
29 \( 1 + 1.24iT - 841T^{2} \)
31 \( 1 + 1.50T + 961T^{2} \)
37 \( 1 + 62.1T + 1.36e3T^{2} \)
41 \( 1 + 55.0iT - 1.68e3T^{2} \)
43 \( 1 + 48.4T + 1.84e3T^{2} \)
47 \( 1 + 61.7iT - 2.20e3T^{2} \)
53 \( 1 + 27.1iT - 2.80e3T^{2} \)
59 \( 1 - 58.0iT - 3.48e3T^{2} \)
61 \( 1 - 7.12T + 3.72e3T^{2} \)
67 \( 1 + 103.T + 4.48e3T^{2} \)
71 \( 1 - 31.8iT - 5.04e3T^{2} \)
73 \( 1 + 47.2T + 5.32e3T^{2} \)
79 \( 1 + 40.9T + 6.24e3T^{2} \)
83 \( 1 - 93.8iT - 6.88e3T^{2} \)
89 \( 1 + 92.6iT - 7.92e3T^{2} \)
97 \( 1 - 156.T + 9.40e3T^{2} \)
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

−9.308445078488375698925986161613, −8.515409687451741190901048605233, −7.29939094093983586115149940240, −6.81616574754092594418321040435, −5.32113235357840634493980009563, −4.58066299466546972319709947725, −3.45257957792831572665057956920, −2.53939252228069485230064096355, −1.66225064873856444956413577674, −0.20295195059078057477052659184, 1.72541225105518808133438636056, 2.99922153231969745959229326844, 4.69092270454957752177281233070, 4.98982339942785234810595666622, 6.04638231973225696539501362599, 6.93456185920236411498653337538, 7.58082039795802328230157839127, 8.291553351885786546061283565825, 9.050725804962360755586555361949, 9.963390210824624083931293791664

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