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Properties

Label 2-33e2-3.2-c2-0-6
Degree $2$
Conductor $1089$
Sign $-0.577 - 0.816i$
Analytic cond. $29.6731$
Root an. cond. $5.44730$
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  + 0.311i·2-s + 3.90·4-s − 5.94i·5-s − 8.67·7-s + 2.46i·8-s + 1.85·10-s − 18.6·13-s − 2.69i·14-s + 14.8·16-s + 9.59i·17-s − 5.76·19-s − 23.2i·20-s + 41.5i·23-s − 10.3·25-s − 5.82i·26-s + ⋯
L(s)  = 1  + 0.155i·2-s + 0.975·4-s − 1.18i·5-s − 1.23·7-s + 0.307i·8-s + 0.185·10-s − 1.43·13-s − 0.192i·14-s + 0.927·16-s + 0.564i·17-s − 0.303·19-s − 1.16i·20-s + 1.80i·23-s − 0.414·25-s − 0.223i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(29.6731\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6308006842\)
\(L(\frac12)\) \(\approx\) \(0.6308006842\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 - 0.311iT - 4T^{2} \)
5 \( 1 + 5.94iT - 25T^{2} \)
7 \( 1 + 8.67T + 49T^{2} \)
13 \( 1 + 18.6T + 169T^{2} \)
17 \( 1 - 9.59iT - 289T^{2} \)
19 \( 1 + 5.76T + 361T^{2} \)
23 \( 1 - 41.5iT - 529T^{2} \)
29 \( 1 + 17.3iT - 841T^{2} \)
31 \( 1 + 13.6T + 961T^{2} \)
37 \( 1 + 7.21T + 1.36e3T^{2} \)
41 \( 1 - 53.1iT - 1.68e3T^{2} \)
43 \( 1 + 43.3T + 1.84e3T^{2} \)
47 \( 1 - 18.8iT - 2.20e3T^{2} \)
53 \( 1 - 54.5iT - 2.80e3T^{2} \)
59 \( 1 - 35.3iT - 3.48e3T^{2} \)
61 \( 1 - 117.T + 3.72e3T^{2} \)
67 \( 1 + 91.5T + 4.48e3T^{2} \)
71 \( 1 + 115. iT - 5.04e3T^{2} \)
73 \( 1 - 52.9T + 5.32e3T^{2} \)
79 \( 1 + 2.04T + 6.24e3T^{2} \)
83 \( 1 + 28.8iT - 6.88e3T^{2} \)
89 \( 1 - 134. iT - 7.92e3T^{2} \)
97 \( 1 + 9.97T + 9.40e3T^{2} \)
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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

−9.748317431224453604179579345642, −9.354615588762327038353897316333, −8.188847274626039771175410584242, −7.45469922307611906457636955774, −6.60467947369676739343299056490, −5.76224296028138893657859622702, −4.94527551678403581774077054026, −3.69104224531873942048306829789, −2.63393802918110670093716388319, −1.42154885705058961842055415158, 0.17297057741204928159720311492, 2.28622003426101014839589859493, 2.81393598041342808944366913281, 3.69983887406827689671101826900, 5.18973110611165255667330085483, 6.36599652906724425044544271377, 6.90297654369651894335643367749, 7.27007392969517236554219555398, 8.569323794693829060576078983655, 9.802146808313507034301835795110

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