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Properties

Label 2-990-33.29-c1-0-6
Degree $2$
Conductor $990$
Sign $0.938 - 0.345i$
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 + (−0.756 + 0.245i)7-s + (0.309 − 0.951i)8-s − i·10-s + (−2.85 − 1.68i)11-s + (−1.56 + 2.15i)13-s + (0.756 + 0.245i)14-s + (−0.809 + 0.587i)16-s + (4.29 − 3.11i)17-s + (7.97 + 2.59i)19-s + (−0.587 + 0.809i)20-s + (1.32 + 3.04i)22-s − 1.65i·23-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (0.262 + 0.361i)5-s + (−0.286 + 0.0929i)7-s + (0.109 − 0.336i)8-s − 0.316i·10-s + (−0.861 − 0.507i)11-s + (−0.435 + 0.598i)13-s + (0.202 + 0.0657i)14-s + (−0.202 + 0.146i)16-s + (1.04 − 0.756i)17-s + (1.83 + 0.594i)19-s + (−0.131 + 0.180i)20-s + (0.281 + 0.648i)22-s − 0.344i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.938 - 0.345i$
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: \((2,\ 990,\ (\ :1/2),\ 0.938 - 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11984 + 0.199693i\)
\(L(\frac12)\) \(\approx\) \(1.11984 + 0.199693i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.85 + 1.68i)T \)
good7 \( 1 + (0.756 - 0.245i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.56 - 2.15i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.29 + 3.11i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-7.97 - 2.59i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.65iT - 23T^{2} \)
29 \( 1 + (-0.0872 - 0.268i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.39 - 3.92i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.08 - 6.42i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.88 - 5.80i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.23iT - 43T^{2} \)
47 \( 1 + (-2.13 - 0.692i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.30 - 5.91i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-9.85 + 3.20i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.32 - 11.4i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.51T + 67T^{2} \)
71 \( 1 + (-5.83 - 8.03i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-11.7 + 3.82i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.46 + 7.52i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.80 + 2.03i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.594iT - 89T^{2} \)
97 \( 1 + (2.68 + 1.94i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.830431826365484185260745054878, −9.588882916058258160961568900746, −8.346237139309870895227701909974, −7.65682370426740893267397676867, −6.82647243436138938194382742907, −5.75013669437012584962031881288, −4.82372010410021074542865627555, −3.27595961911094380116287899347, −2.72616038027224387841134446960, −1.14154710554238558221320481736, 0.76670384606912207738282320011, 2.30065440338782123706019085522, 3.55046028175335560588746479347, 5.12076674128508489014537357260, 5.49074442912764959471336377780, 6.67574843173514007700298453152, 7.67591649644585392187152986392, 8.029933214192641103797013146384, 9.269750925596511756566229588775, 9.858999318423648959186608436335

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