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Properties

Label 2-990-55.9-c1-0-10
Degree $2$
Conductor $990$
Sign $-0.246 - 0.969i$
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  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (1.45 + 1.69i)5-s + (−2.46 + 3.39i)7-s + (0.587 + 0.809i)8-s + (0.858 + 2.06i)10-s + (−1.40 − 3.00i)11-s + (4.63 + 1.50i)13-s + (−3.39 + 2.46i)14-s + (0.309 + 0.951i)16-s + (−0.309 + 0.100i)17-s + (−0.813 + 0.590i)19-s + (0.178 + 2.22i)20-s + (−0.406 − 3.29i)22-s + 4.04i·23-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.404 + 0.293i)4-s + (0.650 + 0.759i)5-s + (−0.932 + 1.28i)7-s + (0.207 + 0.286i)8-s + (0.271 + 0.652i)10-s + (−0.423 − 0.905i)11-s + (1.28 + 0.417i)13-s + (−0.907 + 0.659i)14-s + (0.0772 + 0.237i)16-s + (−0.0750 + 0.0243i)17-s + (−0.186 + 0.135i)19-s + (0.0398 + 0.498i)20-s + (−0.0867 − 0.701i)22-s + 0.843i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.246 - 0.969i$
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: \((2,\ 990,\ (\ :1/2),\ -0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40417 + 1.80510i\)
\(L(\frac12)\) \(\approx\) \(1.40417 + 1.80510i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.45 - 1.69i)T \)
11 \( 1 + (1.40 + 3.00i)T \)
good7 \( 1 + (2.46 - 3.39i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-4.63 - 1.50i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.309 - 0.100i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.813 - 0.590i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.04iT - 23T^{2} \)
29 \( 1 + (6.54 + 4.75i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.29 - 7.07i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.26 + 1.74i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.34 + 4.60i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.63iT - 43T^{2} \)
47 \( 1 + (4.34 + 5.97i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-7.82 - 2.54i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.57 - 6.22i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.71 + 5.28i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 12.2iT - 67T^{2} \)
71 \( 1 + (2.89 + 8.91i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.38 + 7.41i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.25 + 10.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-9.15 + 2.97i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 4.10T + 89T^{2} \)
97 \( 1 + (2.93 + 0.954i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.31735948994205185739836675813, −9.265734189118929815427969929027, −8.720754458528915613095026646202, −7.53938269132840941391679858834, −6.44013437977376131668529294446, −5.95590476540011161184376905053, −5.42106203965944042577901689628, −3.72389716878862784705171746931, −3.05511876875065369009983203248, −2.01568728065038061195014740907, 0.840632839952097601856904615648, 2.21991629043684798925076949192, 3.61325346117948519221965519307, 4.31041036478807841667807981857, 5.35852015430765778267874807492, 6.23813354023296809304825389319, 7.00933122598138711822835819544, 7.990915467796293749577208065817, 9.137157387874287978340167889502, 9.897788494002214449061894122532

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