↑

Properties

Label 2-990-55.9-c1-0-21
Degree $2$
Conductor $990$
Sign $0.392 + 0.919i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (0.434 − 2.19i)5-s + (2.37 − 3.26i)7-s + (−0.587 − 0.809i)8-s + (−1.09 + 1.95i)10-s + (3.30 + 0.248i)11-s + (6.62 + 2.15i)13-s + (−3.26 + 2.37i)14-s + (0.309 + 0.951i)16-s + (3.55 − 1.15i)17-s + (−4.61 + 3.35i)19-s + (1.64 − 1.51i)20-s + (−3.06 − 1.25i)22-s + 5.37i·23-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.404 + 0.293i)4-s + (0.194 − 0.980i)5-s + (0.896 − 1.23i)7-s + (−0.207 − 0.286i)8-s + (−0.344 + 0.617i)10-s + (0.997 + 0.0750i)11-s + (1.83 + 0.596i)13-s + (−0.872 + 0.634i)14-s + (0.0772 + 0.237i)16-s + (0.862 − 0.280i)17-s + (−1.05 + 0.769i)19-s + (0.366 − 0.339i)20-s + (−0.654 − 0.268i)22-s + 1.12i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.392 + 0.919i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.392 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28107 - 0.845940i\)
\(L(\frac12)\) \(\approx\) \(1.28107 - 0.845940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 \)
5 \( 1 + (-0.434 + 2.19i)T \)
11 \( 1 + (-3.30 - 0.248i)T \)
good7 \( 1 + (-2.37 + 3.26i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-6.62 - 2.15i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.55 + 1.15i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.61 - 3.35i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 5.37iT - 23T^{2} \)
29 \( 1 + (-0.759 - 0.551i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.02 + 3.15i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.43 + 3.35i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.94 - 5.76i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.03iT - 43T^{2} \)
47 \( 1 + (-1.85 - 2.55i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.27 - 1.06i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.898 + 0.653i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.18 - 6.71i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.44iT - 67T^{2} \)
71 \( 1 + (-0.579 - 1.78i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.30 - 4.55i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.108 - 0.332i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.04 - 0.665i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 + (8.97 + 2.91i)T + (78.4 + 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.812156337316505516133937277296, −8.898561233360467653033421000622, −8.337461516288669780414599606896, −7.56429363458037394385223731119, −6.53171282998187919852925931821, −5.58751251398619159670411378996, −4.18253652913050821150236693803, −3.80538389376687501124054644131, −1.60567634967053806852753245212, −1.14304775046489861449390925664, 1.41888309715102804544008675270, 2.55529736519354412102684032663, 3.69876707935141485860452658840, 5.16765568622095714618696356992, 6.29780043799122432305706744299, 6.44540393835508769882481504569, 7.88698666622342572083156482642, 8.555206256015973434741390769525, 9.040093923376520732718160755668, 10.25820896685910535100923358071

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