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Properties

Label 2-33e2-9.7-c1-0-74
Degree $2$
Conductor $1089$
Sign $-0.173 + 0.984i$
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  + (0.5 − 0.866i)2-s + (−1.5 + 0.866i)3-s + (0.500 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 1.73i·6-s + (2 − 3.46i)7-s + 3·8-s + (1.5 − 2.59i)9-s − 0.999·10-s + (−1.5 − 0.866i)12-s + (−1 − 1.73i)13-s + (−1.99 − 3.46i)14-s + (1.5 + 0.866i)15-s + (0.500 − 0.866i)16-s − 4·17-s + (−1.5 − 2.59i)18-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.866 + 0.499i)3-s + (0.250 + 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.707i·6-s + (0.755 − 1.30i)7-s + 1.06·8-s + (0.5 − 0.866i)9-s − 0.316·10-s + (−0.433 − 0.249i)12-s + (−0.277 − 0.480i)13-s + (−0.534 − 0.925i)14-s + (0.387 + 0.223i)15-s + (0.125 − 0.216i)16-s − 0.970·17-s + (−0.353 − 0.612i)18-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.173 + 0.984i$
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: \((2,\ 1089,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.452307638\)
\(L(\frac12)\) \(\approx\) \(1.452307638\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
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 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-9.5 + 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.10415904465397542913543204554, −8.845751675703985759968870215611, −7.86301902968550665147629319634, −7.17438753673190165301444768888, −6.22012149996149691185417770290, −4.84872332465053519567303589009, −4.36019256832316784900076968449, −3.71017380830295321975015457814, −2.12704371623891776278186790072, −0.64563431816277849755901893368, 1.59487807486734172475132795992, 2.50204123203757374863124919182, 4.57970630690062027615572962378, 4.99273174447983405552720401135, 6.00767223225028104562166575000, 6.59510235598678484096179378227, 7.25668812088417939252798353338, 8.300417556931604842921731301395, 9.093319561592022991417688087591, 10.43948540498682382987905312962

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