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Properties

Label 2-990-33.29-c1-0-0
Degree $2$
Conductor $990$
Sign $-0.633 - 0.773i$
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 + (−0.587 − 0.809i)5-s + (1.89 − 0.614i)7-s + (0.309 − 0.951i)8-s + i·10-s + (−3.28 + 0.463i)11-s + (−1.89 + 2.60i)13-s + (−1.89 − 0.614i)14-s + (−0.809 + 0.587i)16-s + (−5.26 + 3.82i)17-s + (−1.96 − 0.639i)19-s + (0.587 − 0.809i)20-s + (2.92 + 1.55i)22-s − 3.58i·23-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.262 − 0.361i)5-s + (0.715 − 0.232i)7-s + (0.109 − 0.336i)8-s + 0.316i·10-s + (−0.990 + 0.139i)11-s + (−0.525 + 0.723i)13-s + (−0.505 − 0.164i)14-s + (−0.202 + 0.146i)16-s + (−1.27 + 0.928i)17-s + (−0.451 − 0.146i)19-s + (0.131 − 0.180i)20-s + (0.624 + 0.331i)22-s − 0.746i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.633 - 0.773i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.633 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0791479 + 0.167142i\)
\(L(\frac12)\) \(\approx\) \(0.0791479 + 0.167142i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (3.28 - 0.463i)T \)
good7 \( 1 + (-1.89 + 0.614i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.89 - 2.60i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.26 - 3.82i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.96 + 0.639i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.58iT - 23T^{2} \)
29 \( 1 + (0.406 + 1.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.993 - 0.721i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.08 + 3.35i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.32 + 4.07i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.18iT - 43T^{2} \)
47 \( 1 + (2.18 + 0.709i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.75 - 10.6i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (5.67 - 1.84i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.18 - 1.63i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.59T + 67T^{2} \)
71 \( 1 + (-2.68 - 3.69i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.96 - 2.91i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.40 - 3.31i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.91 - 1.38i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 + (0.221 + 0.160i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.46914907952488029269425042907, −9.379143515750230346867532852616, −8.628649266412545234988175878071, −7.964832986148884133883265851620, −7.17353469643537159383811724418, −6.12142611001111056990866076164, −4.71277674414102143041958865219, −4.25311179534519057450964924826, −2.66469230168384089505631129586, −1.67730229821117943434706111162, 0.096411279857926992624954127712, 2.01327303126373723542480840734, 3.07901067512725907473671144126, 4.68388462318436338650021980631, 5.31838154313638850595513016285, 6.43234566822220879518961924693, 7.36510250126878483476916077313, 7.979026597162739836734898575326, 8.700770635703927748156265874930, 9.670899056802670378448562495484

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