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Properties

Label 2-990-11.3-c1-0-10
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 + (1.30 − 0.951i)13-s + (0.427 + 1.31i)14-s + (−0.809 − 0.587i)16-s + (2 + 1.45i)17-s + (0.5 + 1.53i)19-s + (−0.809 + 0.587i)20-s + (−0.309 − 3.30i)22-s + 1.85·23-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 + (0.363 − 0.263i)13-s + (0.114 + 0.351i)14-s + (−0.202 − 0.146i)16-s + (0.485 + 0.352i)17-s + (0.114 + 0.353i)19-s + (−0.180 + 0.131i)20-s + (−0.0658 − 0.704i)22-s + 0.386·23-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

\(L(1)\) \(\approx\) \(1.07878 - 0.132611i\)
\(L(\frac12)\) \(\approx\) \(1.07878 - 0.132611i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 0.951i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2 - 1.45i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.5 - 1.53i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.85T + 23T^{2} \)
29 \( 1 + (-3 + 9.23i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.85 + 2.80i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.5 + 1.53i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.95 + 6.01i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + (-0.118 - 0.363i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.11 + 2.99i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.82 + 11.7i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.47 + 3.97i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + (11.0 + 8.05i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.14 - 6.60i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.61 - 2.62i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.4 - 7.60i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (-2.38 + 1.73i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.987296416532940930511111516793, −9.095183905982490020523640356618, −8.008604557048020282216154363412, −7.74214032427166571564680309505, −6.71322185986050285528704417280, −5.74005692406741663176309652917, −4.75009046672939771070810920723, −3.79478523169091029378580733307, −2.25866175428314177952078684945, −0.77694677125955800774087479943, 1.07086350593873107655005238719, 2.66771199607006369834813073942, 3.37462668395870001571368036871, 4.71337688273736155761040871940, 5.74413751736051264344094500644, 6.81530038639035049589194510535, 7.64578929512857849781773372206, 8.603942239846104920653691598615, 8.971413527588177308494811752405, 10.18353448164152537255261027374

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