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Properties

Label 2-1143-3.2-c2-0-16
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  − 2.89i·2-s − 4.37·4-s + 0.368i·5-s − 11.4·7-s + 1.09i·8-s + 1.06·10-s + 12.4i·11-s − 14.3·13-s + 33.0i·14-s − 14.3·16-s + 4.03i·17-s + 4.50·19-s − 1.61i·20-s + 36.0·22-s − 24.7i·23-s + ⋯
L(s)  = 1  − 1.44i·2-s − 1.09·4-s + 0.0737i·5-s − 1.63·7-s + 0.136i·8-s + 0.106·10-s + 1.13i·11-s − 1.10·13-s + 2.36i·14-s − 0.896·16-s + 0.237i·17-s + 0.237·19-s − 0.0807i·20-s + 1.64·22-s − 1.07i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.042941130\)
\(L(\frac12)\) \(\approx\) \(1.042941130\)
\(L(2)\) 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 \( 1 + 2.89iT - 4T^{2} \)
5 \( 1 - 0.368iT - 25T^{2} \)
7 \( 1 + 11.4T + 49T^{2} \)
11 \( 1 - 12.4iT - 121T^{2} \)
13 \( 1 + 14.3T + 169T^{2} \)
17 \( 1 - 4.03iT - 289T^{2} \)
19 \( 1 - 4.50T + 361T^{2} \)
23 \( 1 + 24.7iT - 529T^{2} \)
29 \( 1 + 40.2iT - 841T^{2} \)
31 \( 1 + 0.142T + 961T^{2} \)
37 \( 1 - 70.5T + 1.36e3T^{2} \)
41 \( 1 - 72.7iT - 1.68e3T^{2} \)
43 \( 1 - 32.2T + 1.84e3T^{2} \)
47 \( 1 - 29.8iT - 2.20e3T^{2} \)
53 \( 1 + 46.0iT - 2.80e3T^{2} \)
59 \( 1 - 67.6iT - 3.48e3T^{2} \)
61 \( 1 + 40.9T + 3.72e3T^{2} \)
67 \( 1 + 13.7T + 4.48e3T^{2} \)
71 \( 1 + 14.7iT - 5.04e3T^{2} \)
73 \( 1 - 131.T + 5.32e3T^{2} \)
79 \( 1 - 86.0T + 6.24e3T^{2} \)
83 \( 1 - 141. iT - 6.88e3T^{2} \)
89 \( 1 - 136. iT - 7.92e3T^{2} \)
97 \( 1 - 22.3T + 9.40e3T^{2} \)
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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

−9.600108172677885080164942970728, −9.386060828391773869715978121078, −7.916466935857567371705209777399, −6.87358250359973426194394452230, −6.26618752811320425584250599530, −4.74382617091392988491614551382, −4.03823392787426354282130873781, −2.81769261352253069163862371745, −2.42473591793233031540997136733, −0.77656073286199635285717447137, 0.45058353707194892215432978082, 2.68653152232402512894559498895, 3.61116011444647559440509827453, 4.97751849490272943007531893781, 5.72550067043519981771347581727, 6.44938390173208702929934911839, 7.16238996923408073849319869429, 7.80776178341393118338481635633, 9.059571717766310707138208930170, 9.227815668367470895965168997560

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