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Properties

Label 2-33e2-99.85-c0-0-2
Degree $2$
Conductor $1089$
Sign $-0.216 + 0.976i$
Analytic cond. $0.543481$
Root an. cond. $0.737212$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.575 − 1.29i)2-s + (0.978 + 0.207i)3-s + (−0.669 − 0.743i)4-s + (−0.913 + 0.406i)5-s + (0.831 − 1.14i)6-s + (−0.294 − 1.38i)7-s + (0.913 + 0.406i)9-s + 1.41i·10-s + (−0.500 − 0.866i)12-s + (−1.95 − 0.415i)14-s + (−0.978 + 0.207i)15-s + (0.104 − 0.994i)16-s + (1.05 − 0.946i)18-s + (0.913 + 0.406i)20-s − 1.41i·21-s + ⋯
L(s)  = 1  + (0.575 − 1.29i)2-s + (0.978 + 0.207i)3-s + (−0.669 − 0.743i)4-s + (−0.913 + 0.406i)5-s + (0.831 − 1.14i)6-s + (−0.294 − 1.38i)7-s + (0.913 + 0.406i)9-s + 1.41i·10-s + (−0.500 − 0.866i)12-s + (−1.95 − 0.415i)14-s + (−0.978 + 0.207i)15-s + (0.104 − 0.994i)16-s + (1.05 − 0.946i)18-s + (0.913 + 0.406i)20-s − 1.41i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.216 + 0.976i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ -0.216 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.661457420\)
\(L(\frac12)\) \(\approx\) \(1.661457420\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.978 - 0.207i)T \)
11 \( 1 \)
good2 \( 1 + (-0.575 + 1.29i)T + (-0.669 - 0.743i)T^{2} \)
5 \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \)
7 \( 1 + (0.294 + 1.38i)T + (-0.913 + 0.406i)T^{2} \)
13 \( 1 + (0.978 - 0.207i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.294 - 1.38i)T + (-0.913 + 0.406i)T^{2} \)
31 \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.913 - 0.406i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \)
61 \( 1 + (1.40 + 0.147i)T + (0.978 + 0.207i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.669 - 0.743i)T^{2} \)
83 \( 1 + (1.40 + 0.147i)T + (0.978 + 0.207i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \)
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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

−10.05658802605214049037593533980, −9.358663546949343802614499168077, −8.096725407880222243571027609778, −7.46831683073283949944697811559, −6.76074624274113714801741531640, −4.88394255065497867681319442351, −4.07855154903981425223106410577, −3.54346967428513551941942116057, −2.81815202112658576159413782599, −1.38697578060642748414256129423, 2.10278417985452916650104872269, 3.38439138241093233174617534949, 4.31783817903507970963174537084, 5.22399142127878345734191716243, 6.18975417134652901392379484720, 6.97755755403193247699258928069, 7.914054366546055436269536602337, 8.355140624394309363270090957508, 9.022068543570754253671645558284, 9.954790345350468424118392704126

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