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Properties

Label 2-33e2-9.2-c0-0-0
Degree $2$
Conductor $1089$
Sign $0.984 + 0.173i$
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  + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (1.5 − 0.866i)5-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)12-s + (1.5 + 0.866i)15-s + (−0.499 + 0.866i)16-s + (−1.5 − 0.866i)20-s + (1 − 1.73i)25-s − 0.999·27-s + (−0.5 − 0.866i)31-s + 0.999·36-s + 37-s + 1.73i·45-s + (−1.5 − 0.866i)47-s − 0.999·48-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (1.5 − 0.866i)5-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)12-s + (1.5 + 0.866i)15-s + (−0.499 + 0.866i)16-s + (−1.5 − 0.866i)20-s + (1 − 1.73i)25-s − 0.999·27-s + (−0.5 − 0.866i)31-s + 0.999·36-s + 37-s + 1.73i·45-s + (−1.5 − 0.866i)47-s − 0.999·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (848, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.344487155\)
\(L(\frac12)\) \(\approx\) \(1.344487155\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 - 1.73iT - T^{2} \)
59 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.939132962556423288986642152861, −9.184596196807808871983297595491, −8.952908000561139503204809014889, −7.84286549193142732470097982057, −6.24625777462337605502399224789, −5.64355596691664070058038253679, −4.90793633165688835499495849715, −4.17134878020331786274540355984, −2.58106642770042651491156260282, −1.48666360492076691150571402513, 1.78050740651467284916510559322, 2.76577272010863711402446680769, 3.50145038384949733472795913469, 5.01898316164420595680900820225, 6.12336822320508778587964723993, 6.77533165825582423934110441318, 7.57441967524080583468446067987, 8.432524004762199201255146302930, 9.281402225724897303810430602015, 9.806273550964983182110806312035

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