Properties

Label 2-33e2-11.10-c2-0-17
Degree $2$
Conductor $1089$
Sign $-0.975 - 0.219i$
Analytic cond. $29.6731$
Root an. cond. $5.44730$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.65i·2-s − 9.33·4-s − 5.47·5-s − 7.16i·7-s − 19.4i·8-s − 19.9i·10-s − 6.59i·13-s + 26.1·14-s + 33.7·16-s + 18.4i·17-s − 8.42i·19-s + 51.0·20-s + 17.7·23-s + 4.92·25-s + 24.0·26-s + ⋯
L(s)  = 1  + 1.82i·2-s − 2.33·4-s − 1.09·5-s − 1.02i·7-s − 2.43i·8-s − 1.99i·10-s − 0.507i·13-s + 1.86·14-s + 2.10·16-s + 1.08i·17-s − 0.443i·19-s + 2.55·20-s + 0.770·23-s + 0.196·25-s + 0.926·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.975 - 0.219i$
Analytic conductor: \(29.6731\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1),\ -0.975 - 0.219i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8574689896\)
\(L(\frac12)\) \(\approx\) \(0.8574689896\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 - 3.65iT - 4T^{2} \)
5 \( 1 + 5.47T + 25T^{2} \)
7 \( 1 + 7.16iT - 49T^{2} \)
13 \( 1 + 6.59iT - 169T^{2} \)
17 \( 1 - 18.4iT - 289T^{2} \)
19 \( 1 + 8.42iT - 361T^{2} \)
23 \( 1 - 17.7T + 529T^{2} \)
29 \( 1 + 20.0iT - 841T^{2} \)
31 \( 1 - 35.3T + 961T^{2} \)
37 \( 1 + 48.8T + 1.36e3T^{2} \)
41 \( 1 - 31.6iT - 1.68e3T^{2} \)
43 \( 1 - 45.0iT - 1.84e3T^{2} \)
47 \( 1 - 0.728T + 2.20e3T^{2} \)
53 \( 1 + 70.1T + 2.80e3T^{2} \)
59 \( 1 + 24.1T + 3.48e3T^{2} \)
61 \( 1 - 41.0iT - 3.72e3T^{2} \)
67 \( 1 - 96.0T + 4.48e3T^{2} \)
71 \( 1 + 39.5T + 5.04e3T^{2} \)
73 \( 1 - 70.2iT - 5.32e3T^{2} \)
79 \( 1 + 89.3iT - 6.24e3T^{2} \)
83 \( 1 - 26.2iT - 6.88e3T^{2} \)
89 \( 1 - 118.T + 7.92e3T^{2} \)
97 \( 1 + 33.1T + 9.40e3T^{2} \)
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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

−9.842897290504097404272901281835, −8.772359333094454224465163663058, −8.041892231841753056248325834027, −7.62323337066906346019292680182, −6.81033043795542709920731985667, −6.10205465750575116119759213332, −4.90851002952611967251012251680, −4.26692748332980265139223770927, −3.40666890551555543038577113143, −0.792332597461235407280073036737, 0.40633361313443923992498740497, 1.83015682245037311022683633693, 2.88965414865469522475292102755, 3.62600544639881642187284162768, 4.63033974904846481248909726889, 5.36060189464722331244337505867, 6.88172691165201661731565828914, 8.033012773574817747966193028567, 8.844277257189280724812568465076, 9.329521897538520194716250142034

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