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Properties

Label 2-33e2-99.94-c0-0-2
Degree $2$
Conductor $1089$
Sign $-0.184 + 0.982i$
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 + (−0.669 − 0.743i)5-s + (0.669 − 0.743i)9-s + (−0.499 − 0.866i)12-s + (−0.913 − 0.406i)15-s + (−0.978 + 0.207i)16-s + (−0.669 + 0.743i)20-s + (−1 + 1.73i)23-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 − 45-s + (0.104 − 0.994i)47-s + (−0.809 + 0.587i)48-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (−0.669 − 0.743i)5-s + (0.669 − 0.743i)9-s + (−0.499 − 0.866i)12-s + (−0.913 − 0.406i)15-s + (−0.978 + 0.207i)16-s + (−0.669 + 0.743i)20-s + (−1 + 1.73i)23-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 − 45-s + (0.104 − 0.994i)47-s + (−0.809 + 0.587i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.184 + 0.982i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ -0.184 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.191672635\)
\(L(\frac12)\) \(\approx\) \(1.191672635\)
\(L(1)\) 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 \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \)
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 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
61 \( 1 + (-0.913 + 0.406i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 - 2T + T^{2} \)
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

−9.657735437799248908769446061309, −9.043105789916358802030906990043, −8.208256590696849390113097930435, −7.54150282646494319327294631837, −6.51571750124886692282834370469, −5.56029459950534103433310125685, −4.50374501051364827306764548972, −3.68631195970493028714483764478, −2.24054406024181301185342300060, −1.06772997651965484254035321898, 2.39855207873369818293307235635, 3.12126964966874120600157595047, 4.03037914288316472416734418445, 4.65304156882860530551822674371, 6.35768772281671513098212811931, 7.23575972711484773661325367263, 7.980038752840896874881964969545, 8.433977889754987139399714265202, 9.369805127208094457877127685419, 10.26546734889572193350253956570

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