Properties

Label 2-33e2-99.13-c0-0-2
Degree $2$
Conductor $1089$
Sign $0.509 + 0.860i$
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  + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (0.978 + 0.207i)5-s + (−0.978 + 0.207i)9-s + (−0.499 − 0.866i)12-s + (0.104 − 0.994i)15-s + (0.669 − 0.743i)16-s + (0.978 − 0.207i)20-s + (−1 + 1.73i)23-s + (0.309 + 0.951i)27-s + (−0.669 − 0.743i)31-s + (−0.809 + 0.587i)36-s + (0.809 + 0.587i)37-s − 45-s + (−0.913 − 0.406i)47-s + (−0.809 − 0.587i)48-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (0.978 + 0.207i)5-s + (−0.978 + 0.207i)9-s + (−0.499 − 0.866i)12-s + (0.104 − 0.994i)15-s + (0.669 − 0.743i)16-s + (0.978 − 0.207i)20-s + (−1 + 1.73i)23-s + (0.309 + 0.951i)27-s + (−0.669 − 0.743i)31-s + (−0.809 + 0.587i)36-s + (0.809 + 0.587i)37-s − 45-s + (−0.913 − 0.406i)47-s + (−0.809 − 0.587i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.509 + 0.860i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ 0.509 + 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.364350348\)
\(L(\frac12)\) \(\approx\) \(1.364350348\)
\(L(1)\) 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 \( 1 + (-0.913 + 0.406i)T^{2} \)
5 \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \)
7 \( 1 + (0.978 + 0.207i)T^{2} \)
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 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.978 + 0.207i)T^{2} \)
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 \( 1 + (0.104 + 0.994i)T^{2} \)
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 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.913 + 0.406i)T^{2} \)
83 \( 1 + (0.104 + 0.994i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
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

−9.920096301392769656685910385830, −9.295133322817879207975538353022, −7.927224884303985526244481542183, −7.44500132396354676036060858847, −6.31731685746081615550631157806, −6.05796959836757238344242432899, −5.16162035770059068440866088514, −3.32650842818118920857484555597, −2.20736062269649460467375918487, −1.53858145345521626864743837131, 1.96165119242506724347629611573, 2.93731962987430778313204365455, 4.02966983663376212602488140550, 5.08889551325287005680832207245, 6.03314666031010681557838108023, 6.57044861590771195592715728621, 7.86695309386129586545841803634, 8.640386720099906395131882302500, 9.547284561879085812421707308344, 10.21623154176246049447562431229

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