Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.99i·2-s − 4.95·4-s + 7.40i·5-s − 4.26·7-s + 2.85i·8-s + 22.1·10-s − 11.1i·11-s − 7.91·13-s + 12.7i·14-s − 11.2·16-s − 14.3i·17-s + 18.0·19-s − 36.6i·20-s − 33.4·22-s + 33.5i·23-s + ⋯
L(s)  = 1  − 1.49i·2-s − 1.23·4-s + 1.48i·5-s − 0.608·7-s + 0.356i·8-s + 2.21·10-s − 1.01i·11-s − 0.609·13-s + 0.910i·14-s − 0.704·16-s − 0.842i·17-s + 0.952·19-s − 1.83i·20-s − 1.52·22-s + 1.45i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.499278544\)
\(L(\frac12)\) \(\approx\) \(1.499278544\)
\(L(2)\) 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 \( 1 + 2.99iT - 4T^{2} \)
5 \( 1 - 7.40iT - 25T^{2} \)
7 \( 1 + 4.26T + 49T^{2} \)
11 \( 1 + 11.1iT - 121T^{2} \)
13 \( 1 + 7.91T + 169T^{2} \)
17 \( 1 + 14.3iT - 289T^{2} \)
19 \( 1 - 18.0T + 361T^{2} \)
23 \( 1 - 33.5iT - 529T^{2} \)
29 \( 1 - 23.5iT - 841T^{2} \)
31 \( 1 - 43.9T + 961T^{2} \)
37 \( 1 - 2.75T + 1.36e3T^{2} \)
41 \( 1 - 16.3iT - 1.68e3T^{2} \)
43 \( 1 - 48.3T + 1.84e3T^{2} \)
47 \( 1 - 4.06iT - 2.20e3T^{2} \)
53 \( 1 - 74.2iT - 2.80e3T^{2} \)
59 \( 1 + 55.4iT - 3.48e3T^{2} \)
61 \( 1 - 45.3T + 3.72e3T^{2} \)
67 \( 1 - 124.T + 4.48e3T^{2} \)
71 \( 1 + 116. iT - 5.04e3T^{2} \)
73 \( 1 - 93.3T + 5.32e3T^{2} \)
79 \( 1 + 2.92T + 6.24e3T^{2} \)
83 \( 1 + 72.3iT - 6.88e3T^{2} \)
89 \( 1 - 19.0iT - 7.92e3T^{2} \)
97 \( 1 - 18.2T + 9.40e3T^{2} \)
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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

−9.698865640022206838866842017376, −9.239184348437023643249747922595, −7.81160025860486015970644451596, −6.99245657384255261472932787444, −6.18267396928716147158970674117, −4.98716513809282528806806901305, −3.52384724438803910444224202281, −3.16174279775985762844948579495, −2.39210465440827771377052498115, −0.844640069844061941870258468343, 0.63234404710714892878461884103, 2.30344090311794408347376808586, 4.15546437648124963285171801725, 4.79150020252709150412021817220, 5.54151755564813062741736280478, 6.44737800009277035834935850696, 7.19845915848023733802037391629, 8.158096388543582348103743492541, 8.568846845522775706489225657487, 9.592006980527911119550900430843

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