Properties

Label 2-33e2-11.10-c2-0-29
Degree $2$
Conductor $1089$
Sign $-0.904 + 0.426i$
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  − 3.31i·2-s − 7·4-s − 9.38·5-s + 9.89i·7-s + 9.94i·8-s + 31.1i·10-s + 12.7i·13-s + 32.8·14-s + 5.00·16-s − 4.24i·19-s + 65.6·20-s − 28.1·23-s + 63·25-s + 42.2·26-s − 69.2i·28-s − 13.2i·29-s + ⋯
L(s)  = 1  − 1.65i·2-s − 1.75·4-s − 1.87·5-s + 1.41i·7-s + 1.24i·8-s + 3.11i·10-s + 0.979i·13-s + 2.34·14-s + 0.312·16-s − 0.223i·19-s + 3.28·20-s − 1.22·23-s + 2.52·25-s + 1.62·26-s − 2.47i·28-s − 0.457i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.904 + 0.426i$
Analytic conductor: \(29.6731\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1),\ -0.904 + 0.426i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5597176324\)
\(L(\frac12)\) \(\approx\) \(0.5597176324\)
\(L(2)\) 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 \( 1 + 3.31iT - 4T^{2} \)
5 \( 1 + 9.38T + 25T^{2} \)
7 \( 1 - 9.89iT - 49T^{2} \)
13 \( 1 - 12.7iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 + 4.24iT - 361T^{2} \)
23 \( 1 + 28.1T + 529T^{2} \)
29 \( 1 + 13.2iT - 841T^{2} \)
31 \( 1 + 44T + 961T^{2} \)
37 \( 1 - 44T + 1.36e3T^{2} \)
41 \( 1 + 39.7iT - 1.68e3T^{2} \)
43 \( 1 + 29.6iT - 1.84e3T^{2} \)
47 \( 1 + 9.38T + 2.20e3T^{2} \)
53 \( 1 - 65.6T + 2.80e3T^{2} \)
59 \( 1 + 18.7T + 3.48e3T^{2} \)
61 \( 1 - 69.2iT - 3.72e3T^{2} \)
67 \( 1 + 88T + 4.48e3T^{2} \)
71 \( 1 - 46.9T + 5.04e3T^{2} \)
73 \( 1 + 41.0iT - 5.32e3T^{2} \)
79 \( 1 + 12.7iT - 6.24e3T^{2} \)
83 \( 1 - 92.8iT - 6.88e3T^{2} \)
89 \( 1 + 112.T + 7.92e3T^{2} \)
97 \( 1 - 70T + 9.40e3T^{2} \)
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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

−9.255966152370632900748034831697, −8.822811543193578340318585871170, −7.991381119125492635960472096703, −6.97534700776859454940076125510, −5.58698683906488701723096275243, −4.37425898827163000776343733820, −3.89698079789870994027527128323, −2.86673208673362152082037337584, −1.92653863611851370656441160269, −0.29555585704730719186027147404, 0.67601336902735647994752421435, 3.41393027928313446351803696745, 4.13745357914872663926915114863, 4.81728474488088387969969853155, 5.99868530817628027886733231835, 6.99675131041694832433716493231, 7.63107037348670199349080715257, 7.895409914747986776663892539744, 8.677881388159523526484937303099, 9.886506370766587891483509298713

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