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Properties

Label 2-990-99.65-c1-0-24
Degree $2$
Conductor $990$
Sign $-0.00296 + 0.999i$
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 + (−1.19 − 1.25i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (−1.68 + 0.407i)6-s + (1.93 + 1.11i)7-s − 0.999·8-s + (−0.147 + 2.99i)9-s − 0.999i·10-s + (3.16 + 0.989i)11-s + (−0.489 + 1.66i)12-s + (2.70 − 1.55i)13-s + (1.93 − 1.11i)14-s + (−1.66 − 0.489i)15-s + (−0.5 + 0.866i)16-s + 3.53·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.689 − 0.724i)3-s + (−0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.687 + 0.166i)6-s + (0.732 + 0.422i)7-s − 0.353·8-s + (−0.0490 + 0.998i)9-s − 0.316i·10-s + (0.954 + 0.298i)11-s + (−0.141 + 0.479i)12-s + (0.749 − 0.432i)13-s + (0.517 − 0.299i)14-s + (−0.429 − 0.126i)15-s + (−0.125 + 0.216i)16-s + 0.856·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00296 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00296 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.00296 + 0.999i$
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: \((2,\ 990,\ (\ :1/2),\ -0.00296 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33485 - 1.33882i\)
\(L(\frac12)\) \(\approx\) \(1.33485 - 1.33882i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 1.25i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-3.16 - 0.989i)T \)
good7 \( 1 + (-1.93 - 1.11i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (-2.70 + 1.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.53T + 17T^{2} \)
19 \( 1 + 0.649iT - 19T^{2} \)
23 \( 1 + (-1.72 + 0.997i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0531 - 0.0920i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.17 - 7.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.00T + 37T^{2} \)
41 \( 1 + (1.38 + 2.39i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.0 + 5.78i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.60 - 3.23i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.95iT - 53T^{2} \)
59 \( 1 + (2.80 - 1.61i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.4 + 6.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.91 - 8.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.16iT - 71T^{2} \)
73 \( 1 - 1.40iT - 73T^{2} \)
79 \( 1 + (2.59 + 1.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.91 + 8.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.269iT - 89T^{2} \)
97 \( 1 + (2.53 - 4.38i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.06158183800867179038782600477, −8.884471459143364854392610113177, −8.254035302832691467423216846814, −7.05985661969894608739965038297, −6.20446533104585895060524960849, −5.36838718161588868955097881598, −4.70183908475711339903533472316, −3.31499704834220971657968516315, −1.90679455919764487625410850596, −1.11584878819852817593807903733, 1.27794542167151089533246024218, 3.32414938601999858068269154791, 4.16787704013248024035673037762, 4.99121593458467175487537583776, 5.99188821887115291199857048555, 6.48252490145854078282678429855, 7.55790702794523608396241109686, 8.562368563624154475027740990330, 9.392653832206400931467309228298, 10.16417880305191293689977853066

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