↑

Properties

Label 2-33e2-9.4-c1-0-85
Degree $2$
Conductor $1089$
Sign $-0.766 + 0.642i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (1 − 1.73i)4-s + (1.5 − 2.59i)5-s + (−2 − 3.46i)7-s + (1.5 − 2.59i)9-s + 3.46i·12-s + (1 − 1.73i)13-s + 5.19i·15-s + (−1.99 − 3.46i)16-s + 6·17-s − 2·19-s + (−3 − 5.19i)20-s + (6 + 3.46i)21-s + (−1.5 + 2.59i)23-s + (−2 − 3.46i)25-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (0.5 − 0.866i)4-s + (0.670 − 1.16i)5-s + (−0.755 − 1.30i)7-s + (0.5 − 0.866i)9-s + 0.999i·12-s + (0.277 − 0.480i)13-s + 1.34i·15-s + (−0.499 − 0.866i)16-s + 1.45·17-s − 0.458·19-s + (−0.670 − 1.16i)20-s + (1.30 + 0.755i)21-s + (−0.312 + 0.541i)23-s + (−0.400 − 0.692i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.766 + 0.642i$
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: \((2,\ 1089,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.250907722\)
\(L(\frac12)\) \(\approx\) \(1.250907722\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
11 \( 1 \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9 + 15.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.822530052921661474063983715084, −9.147821411631672704018238075976, −7.73451639878917837771893450548, −6.79880729800116557526685028578, −5.89495618512608948460977217050, −5.45146433141669859431636896111, −4.48274017825728164068103373565, −3.43411460564651078515139306897, −1.45549053258398279060303361624, −0.63347067911613628278675938508, 2.02431126083823944821201004719, 2.71117665941221349499080263443, 3.79998674222148296345970343300, 5.51986462890076320052159381974, 6.06870112427556802797557661176, 6.73679393618040105307303390248, 7.42370886448027155827423725067, 8.433566841038936634568112371419, 9.484835208256931200219169708044, 10.31236320986253739819206566823

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