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Properties

Label 2-990-99.65-c1-0-26
Degree $2$
Conductor $990$
Sign $0.737 + 0.675i$
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 + (−0.152 + 1.72i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−1.41 − 0.994i)6-s + (0.0947 + 0.0546i)7-s + 0.999·8-s + (−2.95 − 0.525i)9-s − 0.999i·10-s + (−2.02 − 2.62i)11-s + (1.57 − 0.730i)12-s + (−4.34 + 2.50i)13-s + (−0.0947 + 0.0546i)14-s + (−0.730 − 1.57i)15-s + (−0.5 + 0.866i)16-s + 5.43·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.0879 + 0.996i)3-s + (−0.249 − 0.433i)4-s + (−0.387 + 0.223i)5-s + (−0.578 − 0.406i)6-s + (0.0358 + 0.0206i)7-s + 0.353·8-s + (−0.984 − 0.175i)9-s − 0.316i·10-s + (−0.610 − 0.791i)11-s + (0.453 − 0.210i)12-s + (−1.20 + 0.695i)13-s + (−0.0253 + 0.0146i)14-s + (−0.188 − 0.405i)15-s + (−0.125 + 0.216i)16-s + 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.737 + 0.675i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.737 + 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.368229 - 0.143067i\)
\(L(\frac12)\) \(\approx\) \(0.368229 - 0.143067i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.152 - 1.72i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (2.02 + 2.62i)T \)
good7 \( 1 + (-0.0947 - 0.0546i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (4.34 - 2.50i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.43T + 17T^{2} \)
19 \( 1 + 3.92iT - 19T^{2} \)
23 \( 1 + (2.33 - 1.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.51 + 6.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.17 + 7.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + (-4.65 - 8.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.78 - 2.76i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.66 - 2.11i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.2iT - 53T^{2} \)
59 \( 1 + (-3.93 + 2.27i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.64 + 4.98i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.459 - 0.795i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + 5.22iT - 73T^{2} \)
79 \( 1 + (0.0924 + 0.0533i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.41 + 12.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.300iT - 89T^{2} \)
97 \( 1 + (-1.17 + 2.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.793942768265172969704263674668, −9.185472538319887985094011629799, −8.137288980264106267928519914545, −7.59023599472444450506049516437, −6.43019324730676074522069506745, −5.50365327819833775756032355403, −4.77064164321201824193299401550, −3.72786506774367394202939508879, −2.57615402179778019383564110910, −0.21609437783292882692132199716, 1.31404857513066301115918325312, 2.48479790863401859919852075436, 3.48200466584197615851907992678, 4.94832474006104977527700195033, 5.65707831032970439590962376754, 7.20000793249873293740960170956, 7.51948741250024617689699800153, 8.321615569187293603243673997904, 9.215372430730845113758042483909, 10.38897482491364975034979195223

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