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Properties

Label 2-33e2-9.7-c1-0-92
Degree $2$
Conductor $1089$
Sign $-0.991 - 0.132i$
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  + (1.00 − 1.73i)2-s + (0.413 − 1.68i)3-s + (−1.01 − 1.76i)4-s + (−0.00272 − 0.00472i)5-s + (−2.51 − 2.40i)6-s + (1.91 − 3.31i)7-s − 0.0695·8-s + (−2.65 − 1.39i)9-s − 0.0109·10-s + (−3.38 + 0.982i)12-s + (1.76 + 3.06i)13-s + (−3.84 − 6.65i)14-s + (−0.00906 + 0.00263i)15-s + (1.96 − 3.40i)16-s − 4.57·17-s + (−5.08 + 3.22i)18-s + ⋯
L(s)  = 1  + (0.710 − 1.23i)2-s + (0.238 − 0.971i)3-s + (−0.508 − 0.881i)4-s + (−0.00121 − 0.00211i)5-s + (−1.02 − 0.983i)6-s + (0.723 − 1.25i)7-s − 0.0245·8-s + (−0.886 − 0.463i)9-s − 0.00346·10-s + (−0.976 + 0.283i)12-s + (0.490 + 0.848i)13-s + (−1.02 − 1.77i)14-s + (−0.00234 + 0.000680i)15-s + (0.491 − 0.850i)16-s − 1.10·17-s + (−1.19 + 0.760i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.991 - 0.132i$
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: \((2,\ 1089,\ (\ :1/2),\ -0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.688900995\)
\(L(\frac12)\) \(\approx\) \(2.688900995\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.413 + 1.68i)T \)
11 \( 1 \)
good2 \( 1 + (-1.00 + 1.73i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.00272 + 0.00472i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.91 + 3.31i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (-1.76 - 3.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.57T + 17T^{2} \)
19 \( 1 + 0.684T + 19T^{2} \)
23 \( 1 + (-0.760 - 1.31i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.181 - 0.314i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.14 + 3.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.16T + 37T^{2} \)
41 \( 1 + (-3.33 - 5.78i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.65 - 6.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.31 - 5.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 + (-2.87 - 4.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.77 - 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.80 + 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.78T + 71T^{2} \)
73 \( 1 + 0.0668T + 73T^{2} \)
79 \( 1 + (1.72 - 2.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.84 - 6.65i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.897T + 89T^{2} \)
97 \( 1 + (-7.62 + 13.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.620634531095374399372591526116, −8.599890388690978280748488975950, −7.70460000331979512369983447470, −6.98818052359990877344144203410, −6.03597605794897302233570364103, −4.56939854816876226615293233587, −4.13186610669728355270139846935, −2.89603088782214971600030906282, −1.88148002322686928146958349314, −0.977992162436234587155185865018, 2.21550062441929839247718661537, 3.49260750726563050328398425279, 4.52692241220946259644937528238, 5.30697813341100458092817997971, 5.72301455760793001771090758132, 6.78851855020956003082203160084, 7.87878707045455130349059567634, 8.651524826415823783232594907614, 9.007713444258388202711163959845, 10.36443276047257566969830584698

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