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Properties

Label 2-990-55.4-c1-0-3
Degree $2$
Conductor $990$
Sign $-0.964 + 0.265i$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.450 + 2.19i)5-s + (−2.88 + 0.937i)7-s + (0.951 + 0.309i)8-s + (−2.03 − 0.922i)10-s + (2.76 + 1.83i)11-s + (−1.21 + 1.67i)13-s + (0.937 − 2.88i)14-s + (−0.809 + 0.587i)16-s + (−0.794 − 1.09i)17-s + (0.724 − 2.23i)19-s + (1.94 − 1.10i)20-s + (−3.10 + 1.15i)22-s + 8.76i·23-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.154 − 0.475i)4-s + (0.201 + 0.979i)5-s + (−1.09 + 0.354i)7-s + (0.336 + 0.109i)8-s + (−0.644 − 0.291i)10-s + (0.833 + 0.552i)11-s + (−0.336 + 0.463i)13-s + (0.250 − 0.771i)14-s + (−0.202 + 0.146i)16-s + (−0.192 − 0.265i)17-s + (0.166 − 0.511i)19-s + (0.434 − 0.247i)20-s + (−0.662 + 0.246i)22-s + 1.82i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.964 + 0.265i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0788712 - 0.582600i\)
\(L(\frac12)\) \(\approx\) \(0.0788712 - 0.582600i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 \)
5 \( 1 + (-0.450 - 2.19i)T \)
11 \( 1 + (-2.76 - 1.83i)T \)
good7 \( 1 + (2.88 - 0.937i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.21 - 1.67i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.794 + 1.09i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.724 + 2.23i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.76iT - 23T^{2} \)
29 \( 1 + (2.05 + 6.31i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (7.53 + 5.47i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.93 + 1.27i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.40 - 7.39i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.00iT - 43T^{2} \)
47 \( 1 + (5.35 + 1.73i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.20 - 8.54i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.408 + 1.25i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.07 + 2.95i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 + (8.67 - 6.30i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.93 - 0.627i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.56 + 6.22i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.33 + 10.0i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.64T + 89T^{2} \)
97 \( 1 + (-5.71 + 7.86i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.997001369606386006924372465101, −9.581215240172346320507395784017, −9.084594295419013738391532408769, −7.60469210336986264415789495131, −7.12011205327844186965161405324, −6.29758994328699019709691733122, −5.66179779380850429499359473700, −4.22723419544696813269051328482, −3.14951196074123115176429237103, −1.92216603589076218641350697784, 0.30490289933914588893970617503, 1.61089381488020880575729718021, 3.11795054824962570455320022984, 3.95487510287745970152025642465, 5.05954803851478405099276910689, 6.17747582924957644090892261319, 6.99822891542547935211036831955, 8.194887396521065387198900097709, 8.866093452254120291376488405580, 9.506407061130466846199688184532

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