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Properties

Label 2-990-99.65-c1-0-11
Degree $2$
Conductor $990$
Sign $-0.758 - 0.651i$
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 + (1.07 + 1.35i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−1.71 + 0.254i)6-s + (2.99 + 1.73i)7-s + 0.999·8-s + (−0.678 + 2.92i)9-s − 0.999i·10-s + (−2.44 + 2.23i)11-s + (0.635 − 1.61i)12-s + (4.08 − 2.36i)13-s + (−2.99 + 1.73i)14-s + (−1.61 − 0.635i)15-s + (−0.5 + 0.866i)16-s + 6.28·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.622 + 0.782i)3-s + (−0.249 − 0.433i)4-s + (−0.387 + 0.223i)5-s + (−0.699 + 0.104i)6-s + (1.13 + 0.654i)7-s + 0.353·8-s + (−0.226 + 0.974i)9-s − 0.316i·10-s + (−0.738 + 0.674i)11-s + (0.183 − 0.465i)12-s + (1.13 − 0.654i)13-s + (−0.801 + 0.462i)14-s + (−0.415 − 0.164i)15-s + (−0.125 + 0.216i)16-s + 1.52·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.758 - 0.651i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.758 - 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.572757 + 1.54654i\)
\(L(\frac12)\) \(\approx\) \(0.572757 + 1.54654i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 - 1.35i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (2.44 - 2.23i)T \)
good7 \( 1 + (-2.99 - 1.73i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (-4.08 + 2.36i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.28T + 17T^{2} \)
19 \( 1 - 1.66iT - 19T^{2} \)
23 \( 1 + (2.01 - 1.16i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.90 - 3.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.66 + 2.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.17T + 37T^{2} \)
41 \( 1 + (2.12 + 3.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0168 - 0.00970i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.34 + 3.66i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.5iT - 53T^{2} \)
59 \( 1 + (0.935 - 0.540i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.56 - 1.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.90 - 5.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 + (3.58 + 2.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.30 + 3.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.9iT - 89T^{2} \)
97 \( 1 + (-5.93 + 10.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.24390939477608289367848678633, −9.383569484404641030505921653203, −8.377344487456995067483884796492, −8.035960985731326405680176411135, −7.35717709075055568268683902654, −5.71694472515338483092193529529, −5.28557834123401822619823724199, −4.18044422953588612014564486770, −3.11846238438052052676163994876, −1.72477465594303677666133810499, 0.870287678159676940845380684077, 1.79549225794666297322026573541, 3.19173497029811183752666248260, 3.97820447339133661501851189569, 5.19708964831185257915177594040, 6.44032813662429342278624428918, 7.60850748355394960854927822883, 8.067512702405767788402796656752, 8.585911906485916649890970531740, 9.603498479332510368134332189952

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