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Properties

Label 2-33e2-99.61-c0-0-2
Degree $2$
Conductor $1089$
Sign $-0.996 - 0.0804i$
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  + (0.294 − 1.38i)2-s + (0.104 − 0.994i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)5-s + (−1.34 − 0.437i)6-s + (−1.40 + 0.147i)7-s + (−0.978 − 0.207i)9-s − 1.41i·10-s + (−0.499 + 0.866i)12-s + (−0.209 + 1.98i)14-s + (−0.104 − 0.994i)15-s + (−0.669 − 0.743i)16-s + (−0.575 + 1.29i)18-s + (−0.978 − 0.207i)20-s + 1.41i·21-s + ⋯
L(s)  = 1  + (0.294 − 1.38i)2-s + (0.104 − 0.994i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)5-s + (−1.34 − 0.437i)6-s + (−1.40 + 0.147i)7-s + (−0.978 − 0.207i)9-s − 1.41i·10-s + (−0.499 + 0.866i)12-s + (−0.209 + 1.98i)14-s + (−0.104 − 0.994i)15-s + (−0.669 − 0.743i)16-s + (−0.575 + 1.29i)18-s + (−0.978 − 0.207i)20-s + 1.41i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.996 - 0.0804i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ -0.996 - 0.0804i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.215094054\)
\(L(\frac12)\) \(\approx\) \(1.215094054\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.104 + 0.994i)T \)
11 \( 1 \)
good2 \( 1 + (-0.294 + 1.38i)T + (-0.913 - 0.406i)T^{2} \)
5 \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \)
7 \( 1 + (1.40 - 0.147i)T + (0.978 - 0.207i)T^{2} \)
13 \( 1 + (0.104 + 0.994i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1.40 + 0.147i)T + (0.978 - 0.207i)T^{2} \)
31 \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.978 + 0.207i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \)
61 \( 1 + (-1.05 + 0.946i)T + (0.104 - 0.994i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.913 - 0.406i)T^{2} \)
83 \( 1 + (-1.05 + 0.946i)T + (0.104 - 0.994i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \)
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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

−9.714353812978840469588780155132, −9.255360993730004031164048611441, −8.139014446007905667041791877221, −6.89145858436435959260724100308, −6.29201416496053131364128384547, −5.39748251262291987750017796727, −3.98444549512726115950008617039, −2.84850785346325545104655531480, −2.33830857970292632373847837164, −1.03721026632452037312339256611, 2.53975633314205746964399173467, 3.59960799905316520659475671597, 4.73930342774084517571588061707, 5.51009992266980014430052475859, 6.43634546070287064711098911087, 6.64313405272068389527905242861, 8.035875755882814729459533903350, 8.792662668515614363534157571638, 9.746153939832171650875288206815, 10.06176808409886571986978124142

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