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Properties

Label 2-990-9.4-c1-0-26
Degree $2$
Conductor $990$
Sign $0.173 - 0.984i$
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 + (1.70 + 0.300i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.592 + 1.62i)6-s + (1.93 + 3.35i)7-s − 0.999·8-s + (2.81 + 1.02i)9-s + 0.999·10-s + (0.5 + 0.866i)11-s + (−1.11 + 1.32i)12-s + (1.85 − 3.21i)13-s + (−1.93 + 3.35i)14-s + (1.11 − 1.32i)15-s + (−0.5 − 0.866i)16-s − 4.63·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.984 + 0.173i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (0.241 + 0.664i)6-s + (0.733 + 1.26i)7-s − 0.353·8-s + (0.939 + 0.342i)9-s + 0.316·10-s + (0.150 + 0.261i)11-s + (−0.321 + 0.383i)12-s + (0.515 − 0.892i)13-s + (−0.518 + 0.897i)14-s + (0.287 − 0.342i)15-s + (−0.125 − 0.216i)16-s − 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.173 - 0.984i$
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: \((2,\ 990,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27053 + 1.90520i\)
\(L(\frac12)\) \(\approx\) \(2.27053 + 1.90520i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.300i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-1.93 - 3.35i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-1.85 + 3.21i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.63T + 17T^{2} \)
19 \( 1 - 3.59T + 19T^{2} \)
23 \( 1 + (0.754 - 1.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.85 + 6.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.439 - 0.761i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.16T + 37T^{2} \)
41 \( 1 + (5.02 - 8.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.20 + 3.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.05 - 3.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.59T + 53T^{2} \)
59 \( 1 + (1.04 - 1.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.294 + 0.509i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.86 + 6.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + (-1.90 - 3.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.31 + 10.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (-2.03 - 3.53i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.770673425917215931195346300895, −9.161609293228845437837290144741, −8.346960340569776108621072040330, −7.980840467472106340380579430752, −6.83359404345257627094790119033, −5.69936928093273698034132200949, −5.02881182674891224030881353628, −4.03368888866943080313269806836, −2.85462437104757269863819821070, −1.81178027683286420855293208003, 1.28565600335986524025571631136, 2.22535981689085822806457046944, 3.58204123548798463417496884535, 4.08231339350892057815176433856, 5.17919975563378458208406429702, 6.71849440135263293154027840556, 7.14118935920866256141615044380, 8.283541622284576275772279112383, 9.024294578196671825575149976090, 9.847438911649833181518956312183

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