↑

Properties

Label 2-990-11.4-c1-0-10
Degree $2$
Conductor $990$
Sign $0.642 - 0.766i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (−0.309 + 0.951i)8-s + 10-s + (3.04 + 1.31i)11-s + (1.11 + 0.812i)13-s + (−0.809 + 0.587i)16-s + (−1.30 + 0.951i)17-s + (0.618 − 1.90i)19-s + (0.809 + 0.587i)20-s + (1.69 + 2.85i)22-s + 4.85·23-s + (0.309 − 0.951i)25-s + (0.427 + 1.31i)26-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.109 + 0.336i)8-s + 0.316·10-s + (0.918 + 0.396i)11-s + (0.310 + 0.225i)13-s + (−0.202 + 0.146i)16-s + (−0.317 + 0.230i)17-s + (0.141 − 0.436i)19-s + (0.180 + 0.131i)20-s + (0.360 + 0.608i)22-s + 1.01·23-s + (0.0618 − 0.190i)25-s + (0.0837 + 0.257i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.642 - 0.766i$
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: \((2,\ 990,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25221 + 1.05060i\)
\(L(\frac12)\) \(\approx\) \(2.25221 + 1.05060i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.04 - 1.31i)T \)
good7 \( 1 + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.11 - 0.812i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.30 - 0.951i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.618 + 1.90i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.85T + 23T^{2} \)
29 \( 1 + (-2.04 - 6.29i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.35 - 2.43i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.11 + 6.51i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.14 - 3.52i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.85T + 43T^{2} \)
47 \( 1 + (0.336 - 1.03i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-8.85 - 6.43i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.354 - 1.08i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.23 + 4.53i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + (8.85 - 6.43i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.78 + 7.10i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.618 - 0.449i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + (7.47 + 5.42i)T + (29.9 + 92.2i)T^{2} \)
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

−10.08963418148168602374785554811, −8.970254968767420111117263601741, −8.665694681036309870192591120905, −7.25407263594069839836293448993, −6.74893956463561392743672309671, −5.78886987215884889647588878463, −4.86566336007610893638049635905, −4.03929999493829922005524602986, −2.87313905575613649903829905464, −1.45119008401616600152855906216, 1.14665834434289563582481104869, 2.50918357540366020083343978175, 3.51980692524188889470907718947, 4.47004597974096156049519721191, 5.56283835649289482379076930287, 6.33384742649519275826097155349, 7.09858510615976771371242110813, 8.338912518341349291336864531763, 9.163671975780853956982685755944, 10.03199363793961048408334808790

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