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Properties

Label 2-33e2-9.4-c1-0-61
Degree $2$
Conductor $1089$
Sign $0.919 - 0.391i$
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  + (0.332 + 0.575i)2-s + (1.61 − 0.635i)3-s + (0.779 − 1.34i)4-s + (−0.558 + 0.967i)5-s + (0.900 + 0.715i)6-s + (1.95 + 3.38i)7-s + 2.36·8-s + (2.19 − 2.04i)9-s − 0.741·10-s + (0.397 − 2.67i)12-s + (−1.33 + 2.31i)13-s + (−1.29 + 2.24i)14-s + (−0.284 + 1.91i)15-s + (−0.773 − 1.33i)16-s + 5.54·17-s + (1.90 + 0.580i)18-s + ⋯
L(s)  = 1  + (0.234 + 0.406i)2-s + (0.930 − 0.367i)3-s + (0.389 − 0.674i)4-s + (−0.249 + 0.432i)5-s + (0.367 + 0.292i)6-s + (0.737 + 1.27i)7-s + 0.835·8-s + (0.730 − 0.683i)9-s − 0.234·10-s + (0.114 − 0.770i)12-s + (−0.371 + 0.643i)13-s + (−0.346 + 0.600i)14-s + (−0.0734 + 0.493i)15-s + (−0.193 − 0.334i)16-s + 1.34·17-s + (0.449 + 0.136i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.919 - 0.391i$
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: \((2,\ 1089,\ (\ :1/2),\ 0.919 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.015414031\)
\(L(\frac12)\) \(\approx\) \(3.015414031\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.61 + 0.635i)T \)
11 \( 1 \)
good2 \( 1 + (-0.332 - 0.575i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.558 - 0.967i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.95 - 3.38i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (1.33 - 2.31i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.54T + 17T^{2} \)
19 \( 1 + 5.13T + 19T^{2} \)
23 \( 1 + (1.05 - 1.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.541 + 0.937i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.215 + 0.372i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.70T + 37T^{2} \)
41 \( 1 + (-1.45 + 2.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.11 - 1.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.41 + 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.64T + 53T^{2} \)
59 \( 1 + (-1.18 + 2.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.04 - 3.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.87 + 8.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.31T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 + (2.25 + 3.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.241 + 0.417i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + (4.00 + 6.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.829271899511603955861829730624, −8.949573162293633629077205010846, −8.206168198995498304203407635488, −7.37725683371126204815439902145, −6.67810797861341093775222699580, −5.70908888509807073381478797510, −4.89104912884651217658214950035, −3.60965048599121032900910719592, −2.35938500982069301394731907717, −1.66007335129094912325856219909, 1.33977218196768626284038823891, 2.59919376541546735066752466788, 3.63934835568002479382598763640, 4.27948392822330458214427824814, 5.05300923446737974413234540191, 6.77693727547493454798908333807, 7.68813398775334187509595503369, 8.025732688654043165297990952145, 8.776256703221939880196124258669, 10.17763822114304613244439152783

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