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Properties

Label 2-990-11.4-c1-0-5
Degree $2$
Conductor $990$
Sign $-0.394 - 0.918i$
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 + (0.809 + 2.48i)7-s + (−0.309 + 0.951i)8-s − 10-s + (2.54 − 2.12i)11-s + (1.30 + 0.951i)13-s + (−0.809 + 2.48i)14-s + (−0.809 + 0.587i)16-s + (−4.23 + 3.07i)17-s + (−1.26 + 3.88i)19-s + (−0.809 − 0.587i)20-s + (3.30 − 0.224i)22-s − 0.145·23-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.361 + 0.262i)5-s + (0.305 + 0.941i)7-s + (−0.109 + 0.336i)8-s − 0.316·10-s + (0.767 − 0.641i)11-s + (0.363 + 0.263i)13-s + (−0.216 + 0.665i)14-s + (−0.202 + 0.146i)16-s + (−1.02 + 0.746i)17-s + (−0.289 + 0.892i)19-s + (−0.180 − 0.131i)20-s + (0.705 − 0.0478i)22-s − 0.0304·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.394 - 0.918i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.394 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08920 + 1.65324i\)
\(L(\frac12)\) \(\approx\) \(1.08920 + 1.65324i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.54 + 2.12i)T \)
good7 \( 1 + (-0.809 - 2.48i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.30 - 0.951i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.23 - 3.07i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.26 - 3.88i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.145T + 23T^{2} \)
29 \( 1 + (-0.381 - 1.17i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.85 - 4.25i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.263 + 0.812i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.572 + 1.76i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.23T + 43T^{2} \)
47 \( 1 + (3.5 - 10.7i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.736 - 0.534i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.736 + 2.26i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.23 - 5.25i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.763T + 67T^{2} \)
71 \( 1 + (-10.7 + 7.77i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.527 + 1.62i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.23 - 0.898i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-12.7 + 9.23i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (-9.70 - 7.05i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.42185018162458223655979563220, −9.077966971646245942884300326976, −8.545151239372250503868021675802, −7.80325616394896683522593331056, −6.44671531535544400556376120123, −6.24132083384146868666677256804, −5.02675774167075249182505265353, −4.06659721110152440853356882766, −3.15257291270504479364362388752, −1.81777555606761575232908817199, 0.77597216124597212794817646144, 2.18135969184387473494269605891, 3.55077765298114147027986070881, 4.43090331516394958823757474460, 4.96068994705655169139085054624, 6.44125587295255857430542522984, 7.00187376665692927817341222094, 8.029873168600945726767336305283, 9.007647835703675106482933507162, 9.850323080719404566839989992066

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