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Properties

Label 2-33e2-99.20-c0-0-0
Degree $2$
Conductor $1089$
Sign $0.195 - 0.980i$
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  + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (1.28 + 1.15i)5-s + (0.669 + 0.743i)9-s + (0.499 − 0.866i)12-s + (−0.704 − 1.58i)15-s + (−0.978 − 0.207i)16-s + (−1.28 + 1.15i)20-s + (0.209 + 1.98i)25-s + (−0.309 − 0.951i)27-s + (−0.978 + 0.207i)31-s + (−0.809 + 0.587i)36-s + (−0.809 − 0.587i)37-s + 1.73i·45-s + (1.72 − 0.181i)47-s + (0.809 + 0.587i)48-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (1.28 + 1.15i)5-s + (0.669 + 0.743i)9-s + (0.499 − 0.866i)12-s + (−0.704 − 1.58i)15-s + (−0.978 − 0.207i)16-s + (−1.28 + 1.15i)20-s + (0.209 + 1.98i)25-s + (−0.309 − 0.951i)27-s + (−0.978 + 0.207i)31-s + (−0.809 + 0.587i)36-s + (−0.809 − 0.587i)37-s + 1.73i·45-s + (1.72 − 0.181i)47-s + (0.809 + 0.587i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.195 - 0.980i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (614, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ 0.195 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9114984965\)
\(L(\frac12)\) \(\approx\) \(0.9114984965\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 \)
good2 \( 1 + (0.104 - 0.994i)T^{2} \)
5 \( 1 + (-1.28 - 1.15i)T + (0.104 + 0.994i)T^{2} \)
7 \( 1 + (0.669 - 0.743i)T^{2} \)
13 \( 1 + (0.913 - 0.406i)T^{2} \)
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 \( 1 + (-0.669 + 0.743i)T^{2} \)
31 \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.669 - 0.743i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-1.72 + 0.181i)T + (0.978 - 0.207i)T^{2} \)
53 \( 1 + (-1.64 - 0.535i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-1.72 - 0.181i)T + (0.978 + 0.207i)T^{2} \)
61 \( 1 + (0.913 + 0.406i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (1.64 - 0.535i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.104 + 0.994i)T^{2} \)
83 \( 1 + (-0.913 - 0.406i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \)
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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

−10.44446317605155359036052045084, −9.546960969455555896223183471299, −8.612876939220960817207785293392, −7.27013725293939634680759544803, −7.09587170333300136926300213112, −6.04326488562486573256170866576, −5.39119609558496383572868786277, −4.08114514986119095057910818157, −2.85303899240567756707259389808, −1.91494186143486417241933318361, 0.994137661310882345297260857200, 2.04836672620635485993504276533, 4.08658544071318543970771537905, 5.06839901069847530090165938103, 5.49105861313866008436045145594, 6.15365162170183116864984564595, 7.06388023937619585780740059034, 8.722732309223830847788003681709, 9.191115735651716572773582222577, 10.08268593094645342504273953230

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