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Properties

Label 2-990-55.14-c1-0-10
Degree $2$
Conductor $990$
Sign $-0.435 - 0.900i$
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  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (−1.94 + 1.10i)5-s + (2.88 + 0.937i)7-s + (−0.951 + 0.309i)8-s + (−2.03 − 0.922i)10-s + (2.76 − 1.83i)11-s + (1.21 + 1.67i)13-s + (0.937 + 2.88i)14-s + (−0.809 − 0.587i)16-s + (0.794 − 1.09i)17-s + (0.724 + 2.23i)19-s + (−0.450 − 2.19i)20-s + (3.10 + 1.15i)22-s + 8.76i·23-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−0.154 + 0.475i)4-s + (−0.869 + 0.494i)5-s + (1.09 + 0.354i)7-s + (−0.336 + 0.109i)8-s + (−0.644 − 0.291i)10-s + (0.833 − 0.552i)11-s + (0.336 + 0.463i)13-s + (0.250 + 0.771i)14-s + (−0.202 − 0.146i)16-s + (0.192 − 0.265i)17-s + (0.166 + 0.511i)19-s + (−0.100 − 0.489i)20-s + (0.662 + 0.246i)22-s + 1.82i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.435 - 0.900i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.435 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.968155 + 1.54457i\)
\(L(\frac12)\) \(\approx\) \(0.968155 + 1.54457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 \)
5 \( 1 + (1.94 - 1.10i)T \)
11 \( 1 + (-2.76 + 1.83i)T \)
good7 \( 1 + (-2.88 - 0.937i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.21 - 1.67i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.794 + 1.09i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.724 - 2.23i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 8.76iT - 23T^{2} \)
29 \( 1 + (2.05 - 6.31i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.53 - 5.47i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.93 + 1.27i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.40 + 7.39i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.00iT - 43T^{2} \)
47 \( 1 + (-5.35 + 1.73i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.20 - 8.54i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.408 - 1.25i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.07 - 2.95i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 + (8.67 + 6.30i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.93 - 0.627i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.56 - 6.22i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.33 + 10.0i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 9.64T + 89T^{2} \)
97 \( 1 + (5.71 + 7.86i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.44599952537268812068683353499, −8.907909061328002306554197267104, −8.697724501873662111198604729273, −7.37456455717810973712601653157, −7.21938819541656894910498294442, −5.84799556426234750914544544512, −5.16383334827331450806075580679, −3.93759234196780678997568217961, −3.38195840468365363836145529974, −1.63348079561651505077811136927, 0.800602638210278471631269949398, 2.05429631164788673338507411633, 3.58385310279419995219515042886, 4.38042923219351170331257023329, 4.96179636672493851129635039372, 6.19206943712312231063988619220, 7.31791000014815274733649943866, 8.123580021710355475155672703645, 8.844569596792460036409309440631, 9.824003656956164369344013491653

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