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Properties

Label 2-990-11.3-c1-0-11
Degree $2$
Conductor $990$
Sign $0.970 + 0.242i$
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 + (−0.427 + 1.31i)7-s + (−0.309 − 0.951i)8-s + 10-s + (−1.69 + 2.85i)11-s + (3.92 − 2.85i)13-s + (0.427 + 1.31i)14-s + (−0.809 − 0.587i)16-s + (5.23 + 3.80i)17-s + (−1.5 − 4.61i)19-s + (0.809 − 0.587i)20-s + (0.309 + 3.30i)22-s + 8.61·23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (−0.161 + 0.496i)7-s + (−0.109 − 0.336i)8-s + 0.316·10-s + (−0.509 + 0.860i)11-s + (1.08 − 0.791i)13-s + (0.114 + 0.351i)14-s + (−0.202 − 0.146i)16-s + (1.26 + 0.922i)17-s + (−0.344 − 1.05i)19-s + (0.180 − 0.131i)20-s + (0.0658 + 0.704i)22-s + 1.79·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40664 - 0.295840i\)
\(L(\frac12)\) \(\approx\) \(2.40664 - 0.295840i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 2.85i)T \)
good7 \( 1 + (0.427 - 1.31i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.92 + 2.85i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.23 - 3.80i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.5 + 4.61i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 8.61T + 23T^{2} \)
29 \( 1 + (2.23 - 6.88i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.85 + 4.25i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.972 - 2.99i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.336 - 1.03i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.23T + 43T^{2} \)
47 \( 1 + (3.64 + 11.2i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.5 - 7.69i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.88 + 8.86i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3 - 2.17i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 15.7T + 67T^{2} \)
71 \( 1 + (-6.61 - 4.80i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.618 + 1.90i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.85 + 2.07i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.23 + 3.80i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (-1.61 + 1.17i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.19348473980852661215391039525, −9.246825720023063387322298342253, −8.387547921798834450654837000629, −7.29842584034170862783228177168, −6.36521868899727894506499038264, −5.53187184525338216601142398175, −4.78538462266926571799223046442, −3.43007153591658744281415598712, −2.69852691211289336250183180594, −1.33795065953779212650964314136, 1.16938040210023586152229319762, 2.89523352286403075452594167667, 3.77520598907886227202609143636, 4.86113077297688733616757943236, 5.76367592125149706600540411370, 6.44049559755280382380478007175, 7.45355328094228359511264639222, 8.256918625452361693653691897339, 9.085493119032372939215753319989, 10.01591816866709659283710728097

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