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Properties

Label 2-33e2-9.4-c1-0-66
Degree $2$
Conductor $1089$
Sign $-0.343 + 0.939i$
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  + (−0.447 − 0.774i)2-s + (1.22 − 1.22i)3-s + (0.599 − 1.03i)4-s + (−1.87 + 3.24i)5-s + (−1.49 − 0.401i)6-s + (0.725 + 1.25i)7-s − 2.86·8-s + (0.00384 − 2.99i)9-s + 3.35·10-s + (−0.536 − 2.00i)12-s + (2.87 − 4.98i)13-s + (0.648 − 1.12i)14-s + (1.67 + 6.27i)15-s + (0.0800 + 0.138i)16-s + 4.79·17-s + (−2.32 + 1.33i)18-s + ⋯
L(s)  = 1  + (−0.316 − 0.547i)2-s + (0.707 − 0.706i)3-s + (0.299 − 0.519i)4-s + (−0.838 + 1.45i)5-s + (−0.610 − 0.164i)6-s + (0.274 + 0.474i)7-s − 1.01·8-s + (0.00128 − 0.999i)9-s + 1.06·10-s + (−0.154 − 0.579i)12-s + (0.798 − 1.38i)13-s + (0.173 − 0.300i)14-s + (0.432 + 1.61i)15-s + (0.0200 + 0.0346i)16-s + 1.16·17-s + (−0.548 + 0.315i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.343 + 0.939i$
Analytic conductor: \(8.69570\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1/2),\ -0.343 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.593779754\)
\(L(\frac12)\) \(\approx\) \(1.593779754\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
11 \( 1 \)
good2 \( 1 + (0.447 + 0.774i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.87 - 3.24i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.725 - 1.25i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-2.87 + 4.98i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.79T + 17T^{2} \)
19 \( 1 + 0.702T + 19T^{2} \)
23 \( 1 + (-0.825 + 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.15 + 3.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.65 + 2.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.73T + 37T^{2} \)
41 \( 1 + (-2.12 + 3.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.05 + 3.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.898 + 1.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.15T + 53T^{2} \)
59 \( 1 + (-2.32 + 4.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.27 - 2.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.47 - 7.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.14T + 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 + (0.543 + 0.941i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.90 + 3.29i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.01T + 89T^{2} \)
97 \( 1 + (1.64 + 2.85i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.847940892737250778290344716526, −8.678517200140897208206836982195, −7.933290053643332742055295073589, −7.31006877232471571699854077357, −6.28545145301486784131390917865, −5.70343231565694711700390260661, −3.74600454227617039109102713195, −3.00157130434168555839632115276, −2.30198958535719538328845343279, −0.78421710595864027562396894298, 1.42509527241977210553526380151, 3.17718421914768700812653384631, 4.07234177644725711152434427090, 4.65046080907025378300688837529, 5.85297285414549587129282823070, 7.17589726029773594375798820212, 7.88593148269142333342515846997, 8.415410395276560558485570028439, 9.070663272628981848532849624908, 9.670674838294622853482435102312

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