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Properties

Label 2-990-11.3-c1-0-14
Degree $2$
Conductor $990$
Sign $0.642 + 0.766i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (1 − 3.07i)7-s + (0.309 + 0.951i)8-s − 10-s + (3.04 − 1.31i)11-s + (0.5 − 0.363i)13-s + (1 + 3.07i)14-s + (−0.809 − 0.587i)16-s + (−2.30 − 1.67i)17-s + (−1.61 − 4.97i)19-s + (0.809 − 0.587i)20-s + (−1.69 + 2.85i)22-s − 3.61·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (0.377 − 1.16i)7-s + (0.109 + 0.336i)8-s − 0.316·10-s + (0.918 − 0.396i)11-s + (0.138 − 0.100i)13-s + (0.267 + 0.822i)14-s + (−0.202 − 0.146i)16-s + (−0.560 − 0.406i)17-s + (−0.371 − 1.14i)19-s + (0.180 − 0.131i)20-s + (−0.360 + 0.608i)22-s − 0.754·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11966 - 0.522297i\)
\(L(\frac12)\) \(\approx\) \(1.11966 - 0.522297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-3.04 + 1.31i)T \)
good7 \( 1 + (-1 + 3.07i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.5 + 0.363i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.30 + 1.67i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.61 + 4.97i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.61T + 23T^{2} \)
29 \( 1 + (1.57 - 4.84i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.11 + 2.26i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.263 - 0.812i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.61 + 11.1i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.854T + 43T^{2} \)
47 \( 1 + (3.51 + 10.8i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.381 - 0.277i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.11 - 6.51i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.47 - 6.15i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.32T + 67T^{2} \)
71 \( 1 + (-9.23 - 6.71i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.38 + 4.25i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.54 + 6.20i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.47 + 6.15i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.70T + 89T^{2} \)
97 \( 1 + (14.3 - 10.4i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.846781840423382137552107335194, −8.966578596551977640726767367797, −8.306285454184448274052095767496, −7.07516284199309018843203175694, −6.86204158364278588126138349040, −5.73383944943921329127383506089, −4.62438784460150598798806875358, −3.63122731453020331691704682939, −2.08283433273169261431406864507, −0.71645155606979031608310563040, 1.54574385524083320307623768795, 2.32794026481916982257967622213, 3.75511187534788835255004289722, 4.77941007750575807520015353213, 5.99729141608802362410283786732, 6.57569376567755282403495377863, 8.073861871536829616109173721611, 8.395025469474672526500442420640, 9.467076940502157071552438905304, 9.789375570602826406590765601784

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