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Properties

Label 2-33e2-9.7-c1-0-78
Degree $2$
Conductor $1089$
Sign $-0.581 + 0.813i$
Analytic cond. $8.69570$
Root an. cond. $2.94884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.747 + 1.29i)2-s + (−1.65 + 0.512i)3-s + (−0.118 − 0.205i)4-s + (−1.28 − 2.22i)5-s + (0.573 − 2.52i)6-s + (2.55 − 4.41i)7-s − 2.63·8-s + (2.47 − 1.69i)9-s + 3.84·10-s + (0.302 + 0.279i)12-s + (1.48 + 2.56i)13-s + (3.81 + 6.61i)14-s + (3.26 + 3.02i)15-s + (2.20 − 3.82i)16-s − 3.27·17-s + (0.346 + 4.47i)18-s + ⋯
L(s)  = 1  + (−0.528 + 0.916i)2-s + (−0.955 + 0.295i)3-s + (−0.0594 − 0.102i)4-s + (−0.574 − 0.994i)5-s + (0.234 − 1.03i)6-s + (0.964 − 1.67i)7-s − 0.932·8-s + (0.824 − 0.565i)9-s + 1.21·10-s + (0.0872 + 0.0807i)12-s + (0.410 + 0.711i)13-s + (1.02 + 1.76i)14-s + (0.842 + 0.780i)15-s + (0.552 − 0.956i)16-s − 0.793·17-s + (0.0815 + 1.05i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.581 + 0.813i$
Analytic conductor: \(8.69570\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1/2),\ -0.581 + 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1802183808\)
\(L(\frac12)\) \(\approx\) \(0.1802183808\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.65 - 0.512i)T \)
11 \( 1 \)
good2 \( 1 + (0.747 - 1.29i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.28 + 2.22i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.55 + 4.41i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (-1.48 - 2.56i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.27T + 17T^{2} \)
19 \( 1 + 0.893T + 19T^{2} \)
23 \( 1 + (2.83 + 4.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.28 - 5.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.383 + 0.664i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.88T + 37T^{2} \)
41 \( 1 + (-2.34 - 4.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.58 - 9.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.27 + 2.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.76T + 53T^{2} \)
59 \( 1 + (-0.220 - 0.381i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.442 - 0.766i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.67 - 2.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.34T + 71T^{2} \)
73 \( 1 - 1.06T + 73T^{2} \)
79 \( 1 + (1.12 - 1.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.36 + 4.10i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.03T + 89T^{2} \)
97 \( 1 + (4.82 - 8.36i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.398963108798984660150076941801, −8.513143197560704523589226249975, −7.943885502035415050092094789172, −6.97530362811568110551275473726, −6.55730158048072754111121990642, −5.22766413740170862877420156873, −4.44613665087310819463524301696, −3.85930874117614228273751789976, −1.35613319290745954430610652825, −0.11365044883848627276621602051, 1.73009874448460114991558460054, 2.47722849024500276603406100107, 3.73797229170527053927439042684, 5.28143901325335147953747006466, 5.81430187091122225945706108436, 6.73087521131519271942104184880, 7.80480154391060914497180549541, 8.594071515335228508580511635701, 9.504871372469717118804016981117, 10.56696560763281844923218251842

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