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Properties

Label 2-990-11.4-c1-0-3
Degree $2$
Conductor $990$
Sign $-0.470 - 0.882i$
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 + (−0.536 − 1.65i)7-s + (−0.309 + 0.951i)8-s − 10-s + (−2.35 + 2.33i)11-s + (4.96 + 3.60i)13-s + (0.536 − 1.65i)14-s + (−0.809 + 0.587i)16-s + (−0.791 + 0.575i)17-s + (−1.49 + 4.59i)19-s + (−0.809 − 0.587i)20-s + (−3.27 + 0.500i)22-s − 1.22·23-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.361 + 0.262i)5-s + (−0.202 − 0.624i)7-s + (−0.109 + 0.336i)8-s − 0.316·10-s + (−0.710 + 0.703i)11-s + (1.37 + 0.999i)13-s + (0.143 − 0.441i)14-s + (−0.202 + 0.146i)16-s + (−0.192 + 0.139i)17-s + (−0.342 + 1.05i)19-s + (−0.180 − 0.131i)20-s + (−0.698 + 0.106i)22-s − 0.255·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.470 - 0.882i$
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: \((2,\ 990,\ (\ :1/2),\ -0.470 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.859982 + 1.43260i\)
\(L(\frac12)\) \(\approx\) \(0.859982 + 1.43260i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.35 - 2.33i)T \)
good7 \( 1 + (0.536 + 1.65i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.96 - 3.60i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.791 - 0.575i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.49 - 4.59i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.22T + 23T^{2} \)
29 \( 1 + (-2.60 - 8.02i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.82 + 3.50i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.25 - 6.93i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.08 - 3.32i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.21T + 43T^{2} \)
47 \( 1 + (3.26 - 10.0i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.75 + 5.63i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.08 + 3.35i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-7.37 + 5.35i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.84T + 67T^{2} \)
71 \( 1 + (-3.19 + 2.31i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.73 + 11.5i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.57 + 6.23i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.35 + 3.16i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 5.93T + 89T^{2} \)
97 \( 1 + (-11.1 - 8.07i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.42290373465857989602609031319, −9.400982491288952804431733527314, −8.337360787975057628907932808133, −7.68828132811350779053151210409, −6.72912158626716549557092584707, −6.18299519775400532005998201210, −4.92174541866060349187805791801, −4.05702037490290912932504362725, −3.29782034198410741492812669765, −1.75130504893046050922429523706, 0.63602675348653609567725090253, 2.38939776507885808078214942314, 3.31448836565314470851982966083, 4.28240825894852428544030382075, 5.49945788839817009272702338737, 5.90761278440373393179945063090, 7.10046180169699148416871079301, 8.294134044778752199410287478922, 8.743793076514592142409906377942, 9.833173153616905207464723150741

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