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Properties

Label 2-990-99.32-c1-0-2
Degree $2$
Conductor $990$
Sign $-0.995 + 0.0957i$
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 + (1.07 − 1.35i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (1.71 + 0.254i)6-s + (−2.99 + 1.73i)7-s − 0.999·8-s + (−0.678 − 2.92i)9-s − 0.999i·10-s + (0.713 + 3.23i)11-s + (0.635 + 1.61i)12-s + (−4.08 − 2.36i)13-s + (−2.99 − 1.73i)14-s + (−1.61 + 0.635i)15-s + (−0.5 − 0.866i)16-s − 6.28·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.622 − 0.782i)3-s + (−0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.699 + 0.104i)6-s + (−1.13 + 0.654i)7-s − 0.353·8-s + (−0.226 − 0.974i)9-s − 0.316i·10-s + (0.215 + 0.976i)11-s + (0.183 + 0.465i)12-s + (−1.13 − 0.654i)13-s + (−0.801 − 0.462i)14-s + (−0.415 + 0.164i)15-s + (−0.125 − 0.216i)16-s − 1.52·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.995 + 0.0957i$
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: \((2,\ 990,\ (\ :1/2),\ -0.995 + 0.0957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00855616 - 0.178359i\)
\(L(\frac12)\) \(\approx\) \(0.00855616 - 0.178359i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 + 1.35i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.713 - 3.23i)T \)
good7 \( 1 + (2.99 - 1.73i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (4.08 + 2.36i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.28T + 17T^{2} \)
19 \( 1 - 1.66iT - 19T^{2} \)
23 \( 1 + (2.01 + 1.16i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.90 - 3.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.66 - 2.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.17T + 37T^{2} \)
41 \( 1 + (-2.12 + 3.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0168 - 0.00970i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.34 - 3.66i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 12.5iT - 53T^{2} \)
59 \( 1 + (0.935 + 0.540i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.56 - 1.48i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.90 + 5.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 + (-3.58 + 2.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.30 + 3.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.9iT - 89T^{2} \)
97 \( 1 + (-5.93 - 10.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.08310324814498205675085713310, −9.341613030559743127253579245279, −8.653713181041636617462264725182, −7.79411630834080802465805291554, −6.93052255114852584689036662763, −6.48633543853170771990039522822, −5.32112479142919844316855406418, −4.23088581190660127980044078544, −3.10803174074991487661578867873, −2.18278871312045667322012066822, 0.06109067997320622321108979350, 2.34317801309074045357584097774, 3.20919531644956201671345445807, 4.07017314227697523216446410744, 4.70460129544212585660099714051, 6.10331198154773330924786881148, 6.96485022015347013891906950334, 8.026799273863242191499781298963, 9.136423110159414560818897535543, 9.532202957946576370148286295420

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