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Properties

Label 2-1143-3.2-c2-0-77
Degree $2$
Conductor $1143$
Sign $0.577 - 0.816i$
Analytic cond. $31.1444$
Root an. cond. $5.58072$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.86i·2-s − 10.9·4-s − 9.44i·5-s + 11.3·7-s + 26.8i·8-s − 36.5·10-s − 14.1i·11-s + 10.8·13-s − 43.7i·14-s + 60.1·16-s + 9.53i·17-s − 24.9·19-s + 103. i·20-s − 54.8·22-s + 13.0i·23-s + ⋯
L(s)  = 1  − 1.93i·2-s − 2.73·4-s − 1.88i·5-s + 1.61·7-s + 3.36i·8-s − 3.65·10-s − 1.28i·11-s + 0.831·13-s − 3.12i·14-s + 3.76·16-s + 0.560i·17-s − 1.31·19-s + 5.17i·20-s − 2.49·22-s + 0.567i·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: $0.577 - 0.816i$
Analytic conductor: \(31.1444\)
Root analytic conductor: \(5.58072\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1143} (890, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1143,\ (\ :1),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.215187990\)
\(L(\frac12)\) \(\approx\) \(1.215187990\)
\(L(2)\) 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 \( 1 + 3.86iT - 4T^{2} \)
5 \( 1 + 9.44iT - 25T^{2} \)
7 \( 1 - 11.3T + 49T^{2} \)
11 \( 1 + 14.1iT - 121T^{2} \)
13 \( 1 - 10.8T + 169T^{2} \)
17 \( 1 - 9.53iT - 289T^{2} \)
19 \( 1 + 24.9T + 361T^{2} \)
23 \( 1 - 13.0iT - 529T^{2} \)
29 \( 1 + 18.2iT - 841T^{2} \)
31 \( 1 + 38.5T + 961T^{2} \)
37 \( 1 + 51.8T + 1.36e3T^{2} \)
41 \( 1 - 18.2iT - 1.68e3T^{2} \)
43 \( 1 + 31.3T + 1.84e3T^{2} \)
47 \( 1 + 4.56iT - 2.20e3T^{2} \)
53 \( 1 + 31.5iT - 2.80e3T^{2} \)
59 \( 1 + 62.5iT - 3.48e3T^{2} \)
61 \( 1 - 64.5T + 3.72e3T^{2} \)
67 \( 1 - 74.3T + 4.48e3T^{2} \)
71 \( 1 - 111. iT - 5.04e3T^{2} \)
73 \( 1 + 48.7T + 5.32e3T^{2} \)
79 \( 1 + 63.2T + 6.24e3T^{2} \)
83 \( 1 - 51.0iT - 6.88e3T^{2} \)
89 \( 1 + 19.4iT - 7.92e3T^{2} \)
97 \( 1 - 72.1T + 9.40e3T^{2} \)
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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

−8.800828442759949185721238802261, −8.461295333154287419679063911385, −8.171789080224576965774697586974, −5.66552562314938972942653198076, −5.15998035307385743897033142728, −4.25561256694477662373197941701, −3.67900007628221971302291454682, −1.90422033652841523899794649726, −1.42014583131481870937128615355, −0.38509962167049370877901848119, 1.97426259784364986685104765388, 3.69137110775281414276993504953, 4.55611189350555706750691610138, 5.42662050576112659823565959260, 6.41940306884038169969639430606, 7.05841131193729908573518152749, 7.51661228818563950534083192505, 8.346076578367369876882569482533, 9.111003917604469654903068847471, 10.33360976805686646099099047780

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