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Properties

Label 2-1143-3.2-c2-0-18
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  + 2.71i·2-s − 3.37·4-s + 6.95i·5-s + 4.31·7-s + 1.69i·8-s − 18.8·10-s − 10.6i·11-s + 12.5·13-s + 11.7i·14-s − 18.1·16-s + 32.6i·17-s − 15.7·19-s − 23.4i·20-s + 28.9·22-s − 11.1i·23-s + ⋯
L(s)  = 1  + 1.35i·2-s − 0.844·4-s + 1.39i·5-s + 0.616·7-s + 0.211i·8-s − 1.88·10-s − 0.970i·11-s + 0.962·13-s + 0.837i·14-s − 1.13·16-s + 1.91i·17-s − 0.827·19-s − 1.17i·20-s + 1.31·22-s − 0.486i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.635345128\)
\(L(\frac12)\) \(\approx\) \(1.635345128\)
\(L(2)\) 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 \( 1 - 2.71iT - 4T^{2} \)
5 \( 1 - 6.95iT - 25T^{2} \)
7 \( 1 - 4.31T + 49T^{2} \)
11 \( 1 + 10.6iT - 121T^{2} \)
13 \( 1 - 12.5T + 169T^{2} \)
17 \( 1 - 32.6iT - 289T^{2} \)
19 \( 1 + 15.7T + 361T^{2} \)
23 \( 1 + 11.1iT - 529T^{2} \)
29 \( 1 - 42.0iT - 841T^{2} \)
31 \( 1 + 12.3T + 961T^{2} \)
37 \( 1 + 7.23T + 1.36e3T^{2} \)
41 \( 1 + 3.87iT - 1.68e3T^{2} \)
43 \( 1 + 16.1T + 1.84e3T^{2} \)
47 \( 1 - 2.07iT - 2.20e3T^{2} \)
53 \( 1 - 3.76iT - 2.80e3T^{2} \)
59 \( 1 - 64.3iT - 3.48e3T^{2} \)
61 \( 1 + 52.2T + 3.72e3T^{2} \)
67 \( 1 - 112.T + 4.48e3T^{2} \)
71 \( 1 - 56.2iT - 5.04e3T^{2} \)
73 \( 1 + 25.1T + 5.32e3T^{2} \)
79 \( 1 - 58.5T + 6.24e3T^{2} \)
83 \( 1 + 90.6iT - 6.88e3T^{2} \)
89 \( 1 + 76.9iT - 7.92e3T^{2} \)
97 \( 1 + 41.6T + 9.40e3T^{2} \)
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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

−10.36235670746472898078988165801, −8.723622179469313920907858059109, −8.455116276116148797437164991012, −7.58921960678589559955654187732, −6.62105260660679559776127491443, −6.24368384759746593133626777127, −5.46644499267444938279898639969, −4.14759924669502464800151641932, −3.17893655032877026743831072669, −1.79359316222316187914299974107, 0.48790604747349943841761029860, 1.48967271410150622767558596411, 2.35346278100669030057242615745, 3.73752102087974554679355750983, 4.60848923367291495124610756417, 5.15038867886138407558530447306, 6.54429792357984168860715382622, 7.67564374586207270480849670544, 8.542394927443682465879351780118, 9.396724763174436065396890856555

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