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Properties

Label 2-990-11.3-c1-0-18
Degree $2$
Conductor $990$
Sign $-0.736 + 0.676i$
Analytic cond. $7.90518$
Root an. cond. $2.81161$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (1.57 − 4.85i)7-s + (0.309 + 0.951i)8-s + 10-s + (−1.45 − 2.98i)11-s + (−1.07 + 0.783i)13-s + (1.57 + 4.85i)14-s + (−0.809 − 0.587i)16-s + (0.627 + 0.455i)17-s + (−0.0242 − 0.0746i)19-s + (−0.809 + 0.587i)20-s + (2.92 + 1.55i)22-s − 6.64·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.596 − 1.83i)7-s + (0.109 + 0.336i)8-s + 0.316·10-s + (−0.437 − 0.899i)11-s + (−0.299 + 0.217i)13-s + (0.421 + 1.29i)14-s + (−0.202 − 0.146i)16-s + (0.152 + 0.110i)17-s + (−0.00556 − 0.0171i)19-s + (−0.180 + 0.131i)20-s + (0.624 + 0.332i)22-s − 1.38·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.736 + 0.676i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.224256 - 0.575365i\)
\(L(\frac12)\) \(\approx\) \(0.224256 - 0.575365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 \)
5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (1.45 + 2.98i)T \)
good7 \( 1 + (-1.57 + 4.85i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.07 - 0.783i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.627 - 0.455i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.0242 + 0.0746i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.64T + 23T^{2} \)
29 \( 1 + (-0.623 + 1.92i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.86 - 4.99i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.21 - 3.74i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.67 - 11.3i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.01T + 43T^{2} \)
47 \( 1 + (2.03 + 6.24i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.36 + 6.08i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.64 + 5.06i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.03 - 2.20i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 0.588T + 67T^{2} \)
71 \( 1 + (1.87 + 1.36i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.38 + 10.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.97 - 7.24i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.47 + 4.70i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.67T + 89T^{2} \)
97 \( 1 + (-10.8 + 7.90i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.847530030660410070487641956630, −8.528689609640872771460566026212, −8.023667313852193422322148044417, −7.32095631032593156726084297543, −6.51021527026848835639258951204, −5.31754254084663061362012286989, −4.38626287468644588109348378307, −3.45006812689194733582251717248, −1.60042396996527764791075359336, −0.33594409509570660614355034809, 1.98388723149970907751390985621, 2.57618084259895952143139740261, 3.96111741859660174071006465290, 5.17985896234212839307555519444, 5.92331026781894551310571412213, 7.26030041805187737489702472975, 7.912795227703529616288938615689, 8.731363818931073644521000288378, 9.416055937596888096692028589429, 10.26016022554153315277179593251

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