Properties

Label 2-1143-3.2-c2-0-62
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  − 3.66i·2-s − 9.44·4-s − 0.851i·5-s + 2.76·7-s + 19.9i·8-s − 3.12·10-s − 14.2i·11-s + 7.64·13-s − 10.1i·14-s + 35.3·16-s − 9.98i·17-s + 24.6·19-s + 8.04i·20-s − 52.3·22-s − 26.3i·23-s + ⋯
L(s)  = 1  − 1.83i·2-s − 2.36·4-s − 0.170i·5-s + 0.394·7-s + 2.49i·8-s − 0.312·10-s − 1.29i·11-s + 0.587·13-s − 0.723i·14-s + 2.21·16-s − 0.587i·17-s + 1.29·19-s + 0.402i·20-s − 2.37·22-s − 1.14i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.477559837\)
\(L(\frac12)\) \(\approx\) \(1.477559837\)
\(L(2)\) 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 \( 1 + 3.66iT - 4T^{2} \)
5 \( 1 + 0.851iT - 25T^{2} \)
7 \( 1 - 2.76T + 49T^{2} \)
11 \( 1 + 14.2iT - 121T^{2} \)
13 \( 1 - 7.64T + 169T^{2} \)
17 \( 1 + 9.98iT - 289T^{2} \)
19 \( 1 - 24.6T + 361T^{2} \)
23 \( 1 + 26.3iT - 529T^{2} \)
29 \( 1 + 32.4iT - 841T^{2} \)
31 \( 1 + 4.59T + 961T^{2} \)
37 \( 1 - 8.59T + 1.36e3T^{2} \)
41 \( 1 - 60.9iT - 1.68e3T^{2} \)
43 \( 1 + 34.3T + 1.84e3T^{2} \)
47 \( 1 + 1.72iT - 2.20e3T^{2} \)
53 \( 1 + 15.5iT - 2.80e3T^{2} \)
59 \( 1 + 59.5iT - 3.48e3T^{2} \)
61 \( 1 - 31.0T + 3.72e3T^{2} \)
67 \( 1 + 124.T + 4.48e3T^{2} \)
71 \( 1 - 9.71iT - 5.04e3T^{2} \)
73 \( 1 + 98.4T + 5.32e3T^{2} \)
79 \( 1 - 35.2T + 6.24e3T^{2} \)
83 \( 1 + 50.8iT - 6.88e3T^{2} \)
89 \( 1 - 12.0iT - 7.92e3T^{2} \)
97 \( 1 - 35.5T + 9.40e3T^{2} \)
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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

−9.266661748759436732402024779366, −8.584240215722802647176507871280, −7.900017636502422834928911223436, −6.35419706938450738381516543098, −5.23533237649255956890742910424, −4.48149241700523102520080722499, −3.37416640501124173687982291393, −2.75117320549314855012943549680, −1.37178939908071790019223529041, −0.51608011874135387283034464737, 1.43405273927625639076656389428, 3.41896541792469210042467334146, 4.50667739571516500359580495031, 5.24819323216803336743832779569, 5.98843494034554214995305172793, 7.18927494765570746488194388503, 7.26165614644100348185717129537, 8.329322815911282308152860112879, 9.047678180527779276712740660452, 9.782180018406466204379144074987

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