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Properties

Label 2-990-11.5-c1-0-5
Degree $2$
Conductor $990$
Sign $-0.138 - 0.990i$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (−1.27 + 0.929i)7-s + (−0.809 − 0.587i)8-s + 0.999·10-s + (2.99 + 1.42i)11-s + (1.77 + 5.47i)13-s + (−1.27 − 0.929i)14-s + (0.309 − 0.951i)16-s + (2.21 − 6.82i)17-s + (1.06 + 0.776i)19-s + (0.309 + 0.951i)20-s + (−0.427 + 3.28i)22-s + 0.358·23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (−0.483 + 0.351i)7-s + (−0.286 − 0.207i)8-s + 0.316·10-s + (0.903 + 0.428i)11-s + (0.493 + 1.51i)13-s + (−0.341 − 0.248i)14-s + (0.0772 − 0.237i)16-s + (0.537 − 1.65i)17-s + (0.245 + 0.178i)19-s + (0.0690 + 0.212i)20-s + (−0.0910 + 0.701i)22-s + 0.0748·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.138 - 0.990i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.138 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08586 + 1.24870i\)
\(L(\frac12)\) \(\approx\) \(1.08586 + 1.24870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 \)
5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-2.99 - 1.42i)T \)
good7 \( 1 + (1.27 - 0.929i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.77 - 5.47i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.21 + 6.82i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.06 - 0.776i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.358T + 23T^{2} \)
29 \( 1 + (7.82 - 5.68i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.22 - 6.84i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.87 - 5.72i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-9.71 - 7.05i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.41T + 43T^{2} \)
47 \( 1 + (-5.27 - 3.83i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.724 + 2.22i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.35 + 6.06i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.74 - 5.35i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 5.07T + 67T^{2} \)
71 \( 1 + (2.81 - 8.66i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.61 + 4.08i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.56 + 10.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.47 + 7.60i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 7.16T + 89T^{2} \)
97 \( 1 + (0.594 + 1.83i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.806129752465167601913454257468, −9.188823923848458264490518414754, −8.807785147797888496965342366938, −7.43294182493475998017158054417, −6.85861157675800301294538310770, −6.00041756032886699159696198699, −5.03017144582565668609600010586, −4.19093610529383312865968674489, −3.07339831636944751895372191764, −1.43927095287575040073416935463, 0.791831223147882208031799276590, 2.27784674756547067172348955890, 3.63883795788443878625631172168, 3.87548924853976319861819465357, 5.74307393226277073489913626017, 5.93235326796510814337514846151, 7.27382577532482276609976138532, 8.167477356540006746803160817846, 9.121483027545883834861914179380, 9.951797762603350836269080648582

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