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Properties

Label 2-990-99.65-c1-0-37
Degree $2$
Conductor $990$
Sign $-0.0864 + 0.996i$
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 + (1.50 − 0.861i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.00548 − 1.73i)6-s + (1.26 + 0.731i)7-s − 0.999·8-s + (1.51 − 2.58i)9-s + 0.999i·10-s + (3.30 − 0.312i)11-s + (−1.49 − 0.870i)12-s + (0.354 − 0.204i)13-s + (1.26 − 0.731i)14-s + (−0.870 + 1.49i)15-s + (−0.5 + 0.866i)16-s − 2.86·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.867 − 0.497i)3-s + (−0.249 − 0.433i)4-s + (−0.387 + 0.223i)5-s + (0.00224 − 0.707i)6-s + (0.478 + 0.276i)7-s − 0.353·8-s + (0.505 − 0.862i)9-s + 0.316i·10-s + (0.995 − 0.0941i)11-s + (−0.432 − 0.251i)12-s + (0.0981 − 0.0566i)13-s + (0.338 − 0.195i)14-s + (−0.224 + 0.386i)15-s + (−0.125 + 0.216i)16-s − 0.695·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0864 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.0864 + 0.996i$
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: \((2,\ 990,\ (\ :1/2),\ -0.0864 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73147 - 1.88828i\)
\(L(\frac12)\) \(\approx\) \(1.73147 - 1.88828i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 + 0.861i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-3.30 + 0.312i)T \)
good7 \( 1 + (-1.26 - 0.731i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (-0.354 + 0.204i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.86T + 17T^{2} \)
19 \( 1 + 6.41iT - 19T^{2} \)
23 \( 1 + (-7.54 + 4.35i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.07 - 8.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.478 - 0.828i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.03T + 37T^{2} \)
41 \( 1 + (0.185 + 0.322i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.17 + 4.14i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.43 + 3.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.52iT - 53T^{2} \)
59 \( 1 + (-3.76 + 2.17i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.21 - 4.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.265 + 0.459i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.71iT - 71T^{2} \)
73 \( 1 - 3.68iT - 73T^{2} \)
79 \( 1 + (3.20 + 1.85i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.46 - 11.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.53iT - 89T^{2} \)
97 \( 1 + (9.30 - 16.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.587402250246294617762987973426, −8.873909640860017022450063905080, −8.419375511399252269900734131967, −6.99742664959055333726506674431, −6.72240830339112284357517780320, −5.16845480470931544340014925670, −4.22958329453580053026996736397, −3.24413577247018888095221245250, −2.35550910558272424567247174911, −1.09025584333767991508495958058, 1.67004354891977905892108646253, 3.25602515918629668768216821191, 4.09674542765698269955484295998, 4.70828731490304046842877339092, 5.88511262118611845925051468616, 6.98797394677475668076452763452, 7.83007734442152110830072725184, 8.364017080674588711746576378459, 9.315955044391181835298291493449, 9.850471384625083670485470640104

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