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Properties

Label 2-990-11.4-c1-0-19
Degree $2$
Conductor $990$
Sign $-0.642 + 0.766i$
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 + (−1 − 3.07i)7-s + (−0.309 + 0.951i)8-s − 10-s + (−3.04 − 1.31i)11-s + (−2.73 − 1.98i)13-s + (1 − 3.07i)14-s + (−0.809 + 0.587i)16-s + (−4.92 + 3.57i)17-s + (0.381 − 1.17i)19-s + (−0.809 − 0.587i)20-s + (−1.69 − 2.85i)22-s − 6.85·23-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.361 + 0.262i)5-s + (−0.377 − 1.16i)7-s + (−0.109 + 0.336i)8-s − 0.316·10-s + (−0.918 − 0.396i)11-s + (−0.758 − 0.551i)13-s + (0.267 − 0.822i)14-s + (−0.202 + 0.146i)16-s + (−1.19 + 0.868i)17-s + (0.0876 − 0.269i)19-s + (−0.180 − 0.131i)20-s + (−0.360 − 0.608i)22-s − 1.42·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.164537 - 0.352722i\)
\(L(\frac12)\) \(\approx\) \(0.164537 - 0.352722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.04 + 1.31i)T \)
good7 \( 1 + (1 + 3.07i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.73 + 1.98i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.92 - 3.57i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.381 + 1.17i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.85T + 23T^{2} \)
29 \( 1 + (-0.809 - 2.48i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.11 - 0.812i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.97 + 6.06i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.381 - 1.17i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.0901T + 43T^{2} \)
47 \( 1 + (-1.33 + 4.11i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.85 - 4.97i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.35 + 13.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 5.09T + 67T^{2} \)
71 \( 1 + (-8.47 + 6.15i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.14 - 12.7i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.30 - 2.40i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.23 + 2.35i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 + (15.0 + 10.9i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.895964693268847718010512647722, −8.571795893588855149538935638654, −7.82321287753406230729641005743, −7.13458513392420455315312102854, −6.34578169460306173373654861594, −5.31565561944020148422548212535, −4.29105789478820072525649009665, −3.54432021940531699215982221097, −2.37550201068048897114303723931, −0.13289396607306210171495828909, 2.12950577741699391296448564696, 2.78216538830792372434486792709, 4.20039211515647385522598534765, 4.95875296838997163809502297769, 5.82299720815927701313603713092, 6.78102286746600749740277934197, 7.78229039671671130199758523829, 8.759472731249777928947856854008, 9.561952248899567749473356627766, 10.23852321881425447444615618623

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