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Properties

Label 2-990-11.4-c1-0-11
Degree $2$
Conductor $990$
Sign $0.470 + 0.882i$
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 + (−0.536 − 1.65i)7-s + (0.309 − 0.951i)8-s − 10-s + (2.35 − 2.33i)11-s + (4.96 + 3.60i)13-s + (−0.536 + 1.65i)14-s + (−0.809 + 0.587i)16-s + (0.791 − 0.575i)17-s + (−1.49 + 4.59i)19-s + (0.809 + 0.587i)20-s + (−3.27 + 0.500i)22-s + 1.22·23-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.202 − 0.624i)7-s + (0.109 − 0.336i)8-s − 0.316·10-s + (0.710 − 0.703i)11-s + (1.37 + 0.999i)13-s + (−0.143 + 0.441i)14-s + (−0.202 + 0.146i)16-s + (0.192 − 0.139i)17-s + (−0.342 + 1.05i)19-s + (0.180 + 0.131i)20-s + (−0.698 + 0.106i)22-s + 0.255·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.470 + 0.882i$
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: \((2,\ 990,\ (\ :1/2),\ 0.470 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15918 - 0.695851i\)
\(L(\frac12)\) \(\approx\) \(1.15918 - 0.695851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.35 + 2.33i)T \)
good7 \( 1 + (0.536 + 1.65i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.96 - 3.60i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.791 + 0.575i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.49 - 4.59i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.22T + 23T^{2} \)
29 \( 1 + (2.60 + 8.02i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.82 + 3.50i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.25 - 6.93i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.08 + 3.32i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.21T + 43T^{2} \)
47 \( 1 + (-3.26 + 10.0i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.75 - 5.63i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.08 - 3.35i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-7.37 + 5.35i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.84T + 67T^{2} \)
71 \( 1 + (3.19 - 2.31i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.73 + 11.5i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.57 + 6.23i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.35 - 3.16i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 5.93T + 89T^{2} \)
97 \( 1 + (-11.1 - 8.07i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.848255222586911375893613690383, −8.998105168549562703253213013246, −8.464194383107932989887470580146, −7.42898471626770621674772438497, −6.43087785493902717876123662195, −5.77075496100537596934094538260, −4.10667389332238188880246553290, −3.65058702586808438850541064025, −2.02685293597240099692729087800, −0.917502959655405859427447297451, 1.24708326257754889636649539720, 2.59306257154271006795834797330, 3.81297121425925859663033514869, 5.23263347623790557888846924725, 5.94971840198539584389718792355, 6.79277282157923525209288805289, 7.54730063219226241724330517369, 8.854325721376326288117519990751, 8.952484569516118183702901064680, 10.04726722740854081962598993795

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