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Properties

Label 2-33e2-99.47-c0-0-0
Degree $2$
Conductor $1089$
Sign $0.913 + 0.406i$
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  + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (−1.72 − 0.181i)5-s + (−0.104 − 0.994i)9-s + (0.5 − 0.866i)12-s + (1.28 − 1.15i)15-s + (0.913 − 0.406i)16-s + (1.72 − 0.181i)20-s + (1.95 + 0.415i)25-s + (0.809 + 0.587i)27-s + (0.913 + 0.406i)31-s + (0.309 + 0.951i)36-s + (0.309 − 0.951i)37-s + 1.73i·45-s + (0.360 − 1.69i)47-s + (−0.309 + 0.951i)48-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (−1.72 − 0.181i)5-s + (−0.104 − 0.994i)9-s + (0.5 − 0.866i)12-s + (1.28 − 1.15i)15-s + (0.913 − 0.406i)16-s + (1.72 − 0.181i)20-s + (1.95 + 0.415i)25-s + (0.809 + 0.587i)27-s + (0.913 + 0.406i)31-s + (0.309 + 0.951i)36-s + (0.309 − 0.951i)37-s + 1.73i·45-s + (0.360 − 1.69i)47-s + (−0.309 + 0.951i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.913 + 0.406i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ 0.913 + 0.406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3635656276\)
\(L(\frac12)\) \(\approx\) \(0.3635656276\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.669 - 0.743i)T \)
11 \( 1 \)
good2 \( 1 + (0.978 - 0.207i)T^{2} \)
5 \( 1 + (1.72 + 0.181i)T + (0.978 + 0.207i)T^{2} \)
7 \( 1 + (-0.104 + 0.994i)T^{2} \)
13 \( 1 + (0.669 + 0.743i)T^{2} \)
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 \( 1 + (0.104 - 0.994i)T^{2} \)
31 \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.104 + 0.994i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.360 + 1.69i)T + (-0.913 - 0.406i)T^{2} \)
53 \( 1 + (1.01 + 1.40i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.360 - 1.69i)T + (-0.913 + 0.406i)T^{2} \)
61 \( 1 + (0.669 - 0.743i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.01 + 1.40i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.978 + 0.207i)T^{2} \)
83 \( 1 + (-0.669 + 0.743i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
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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

−10.06020877222711465274330322461, −9.094337691612032780875889808841, −8.465960088777360389020179576863, −7.69963777593424874400648462560, −6.70676376674748128068529941457, −5.39683063960763899127126602799, −4.66703828281066826279713335114, −3.95840125991140156828474424085, −3.31919641074968655133768504355, −0.51327809106746081110577102323, 0.990170039254317290133442846089, 2.99356484526563916301213786187, 4.22374297474166463179812140824, 4.76203928270803249437902214337, 5.91608957379477845132158476649, 6.84993963502090522039560678051, 7.900531962492548836387086529612, 8.077287584725451346216435465038, 9.186002337863880186572645181543, 10.28095163521730226145548995752

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