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Properties

Label 2-990-99.32-c1-0-5
Degree $2$
Conductor $990$
Sign $-0.994 - 0.106i$
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  + (−0.5 − 0.866i)2-s + (0.713 + 1.57i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.5i)5-s + (1.01 − 1.40i)6-s + (−3.86 + 2.23i)7-s + 0.999·8-s + (−1.98 + 2.25i)9-s − 0.999i·10-s + (−2.72 − 1.88i)11-s + (−1.72 − 0.171i)12-s + (5.39 + 3.11i)13-s + (3.86 + 2.23i)14-s + (−0.171 + 1.72i)15-s + (−0.5 − 0.866i)16-s − 6.51·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.411 + 0.911i)3-s + (−0.249 + 0.433i)4-s + (0.387 + 0.223i)5-s + (0.412 − 0.574i)6-s + (−1.46 + 0.844i)7-s + 0.353·8-s + (−0.660 + 0.750i)9-s − 0.316i·10-s + (−0.821 − 0.569i)11-s + (−0.497 − 0.0495i)12-s + (1.49 + 0.863i)13-s + (1.03 + 0.597i)14-s + (−0.0442 + 0.445i)15-s + (−0.125 − 0.216i)16-s − 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.994 - 0.106i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.994 - 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0219151 + 0.411401i\)
\(L(\frac12)\) \(\approx\) \(0.0219151 + 0.411401i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.713 - 1.57i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (2.72 + 1.88i)T \)
good7 \( 1 + (3.86 - 2.23i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (-5.39 - 3.11i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.51T + 17T^{2} \)
19 \( 1 + 7.19iT - 19T^{2} \)
23 \( 1 + (-0.160 - 0.0927i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.39 + 2.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.41 - 5.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.21T + 37T^{2} \)
41 \( 1 + (1.73 - 3.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.68 - 1.54i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.56 + 5.51i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.08iT - 53T^{2} \)
59 \( 1 + (11.5 + 6.64i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.19 - 1.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.67 - 9.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.30iT - 71T^{2} \)
73 \( 1 - 5.48iT - 73T^{2} \)
79 \( 1 + (8.26 - 4.77i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.06 - 3.57i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.3iT - 89T^{2} \)
97 \( 1 + (2.61 + 4.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.39603991286139717093375621376, −9.392099504253179410863116964876, −8.943544591201322695731450214511, −8.533872169734871168633036494481, −6.90603563832291723033568503189, −6.13335800070610434304093434011, −5.06377364201513312407457175354, −3.86296905611828655626802673757, −2.99803295188133601494813276994, −2.28164461366235481843535389096, 0.18990601231401115846677712674, 1.66511801316852484243736707463, 3.10643626770789830416256216440, 4.12417516812999620010276085366, 5.85611549657863111055557075720, 6.15011515277119129488161735671, 7.15894431013289990018452347831, 7.76397466507979927974976586670, 8.715605038974522910179975068478, 9.358583611327960542438902150411

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