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Properties

Label 2-990-99.32-c1-0-16
Degree $2$
Conductor $990$
Sign $0.253 - 0.967i$
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 + (1.65 + 0.520i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.375 − 1.69i)6-s + (−3.61 + 2.08i)7-s + 0.999·8-s + (2.45 + 1.72i)9-s − 0.999i·10-s + (2.14 + 2.52i)11-s + (−1.27 + 1.17i)12-s + (−3.39 − 1.96i)13-s + (3.61 + 2.08i)14-s + (1.17 + 1.27i)15-s + (−0.5 − 0.866i)16-s + 1.14·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.953 + 0.300i)3-s + (−0.249 + 0.433i)4-s + (0.387 + 0.223i)5-s + (−0.153 − 0.690i)6-s + (−1.36 + 0.789i)7-s + 0.353·8-s + (0.819 + 0.573i)9-s − 0.316i·10-s + (0.646 + 0.762i)11-s + (−0.368 + 0.337i)12-s + (−0.942 − 0.544i)13-s + (0.966 + 0.558i)14-s + (0.302 + 0.329i)15-s + (−0.125 − 0.216i)16-s + 0.277·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.253 - 0.967i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.253 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10087 + 0.849273i\)
\(L(\frac12)\) \(\approx\) \(1.10087 + 0.849273i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 - 0.520i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-2.14 - 2.52i)T \)
good7 \( 1 + (3.61 - 2.08i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (3.39 + 1.96i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.14T + 17T^{2} \)
19 \( 1 - 4.10iT - 19T^{2} \)
23 \( 1 + (5.50 + 3.17i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.62 - 2.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.55 - 7.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.54T + 37T^{2} \)
41 \( 1 + (3.46 - 5.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.59 + 4.96i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.36 - 4.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.65iT - 53T^{2} \)
59 \( 1 + (2.56 + 1.48i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.80 + 3.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.70 - 8.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.97iT - 71T^{2} \)
73 \( 1 + 15.4iT - 73T^{2} \)
79 \( 1 + (-3.43 + 1.98i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.80 - 4.86i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.0iT - 89T^{2} \)
97 \( 1 + (1.13 + 1.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.00489097913312843186919417315, −9.480020391967570644795957262428, −8.822976664620891560673054617017, −7.83703471054915224716480276106, −6.94369052534865198899840419332, −5.93380629656811804200637267909, −4.64707689470876921843388014885, −3.49919726927475933370819128980, −2.79012073555855266538665914966, −1.84298729634983618394735513151, 0.62027742404174187744063869221, 2.22968685949326907390807727197, 3.50767644539185031167893391458, 4.33510975433274637691185738990, 5.85690993679240928656681596604, 6.61596995958984341728232710304, 7.27920948222678886123183013693, 8.083099278571582307567740635893, 9.165704925586675986255114046853, 9.573629468148667970270063121768

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