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Properties

Label 2-990-9.4-c1-0-16
Degree $2$
Conductor $990$
Sign $0.159 + 0.987i$
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  + (−0.5 − 0.866i)2-s + (−1.73 − 0.0120i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.855 + 1.50i)6-s + (−1.08 − 1.88i)7-s + 0.999·8-s + (2.99 + 0.0418i)9-s + 0.999·10-s + (−0.5 − 0.866i)11-s + (0.876 − 1.49i)12-s + (−2.24 + 3.88i)13-s + (−1.08 + 1.88i)14-s + (0.876 − 1.49i)15-s + (−0.5 − 0.866i)16-s + 0.0509·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.999 − 0.00697i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.349 + 0.614i)6-s + (−0.410 − 0.711i)7-s + 0.353·8-s + (0.999 + 0.0139i)9-s + 0.316·10-s + (−0.150 − 0.261i)11-s + (0.253 − 0.431i)12-s + (−0.621 + 1.07i)13-s + (−0.290 + 0.503i)14-s + (0.226 − 0.385i)15-s + (−0.125 − 0.216i)16-s + 0.0123·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.159 + 0.987i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (661, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.494886 - 0.421179i\)
\(L(\frac12)\) \(\approx\) \(0.494886 - 0.421179i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (1.73 + 0.0120i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (1.08 + 1.88i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (2.24 - 3.88i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.0509T + 17T^{2} \)
19 \( 1 - 2.99T + 19T^{2} \)
23 \( 1 + (3.26 - 5.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.582 - 1.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.35 + 5.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.52T + 37T^{2} \)
41 \( 1 + (-4.51 + 7.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.11 + 3.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.33 + 5.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.99T + 53T^{2} \)
59 \( 1 + (-2.60 + 4.50i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0219 - 0.0379i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.77 + 9.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.04T + 71T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 + (2.34 + 4.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.750 + 1.30i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + (3.56 + 6.18i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.804223945738695809227395196601, −9.488942819822615627705556130935, −7.976912954565684644675354375399, −7.25442496406085705964924809909, −6.53296997339859520208109614283, −5.43538309875365127037538438151, −4.31406216017449095853767782653, −3.56434165604066678161512664777, −2.04802039627080503711816057335, −0.53209633579591193778439088034, 0.909849624741553107231530988709, 2.72484191885662283956784110637, 4.38782826978583176748634007990, 5.14836982085004104850142183037, 5.93173084023331533685139989944, 6.65432007157118101250860974906, 7.70570374798090633616481202909, 8.314729836089616024484835025838, 9.569681614536664958972644614487, 9.935941784268711349055928947754

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