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Properties

Label 2-990-11.5-c1-0-19
Degree $2$
Conductor $990$
Sign $0.387 + 0.922i$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (2.58 − 1.88i)7-s + (−0.809 − 0.587i)8-s + 0.999·10-s + (−3.26 − 0.610i)11-s + (−2.08 − 6.42i)13-s + (2.58 + 1.88i)14-s + (0.309 − 0.951i)16-s + (−0.172 + 0.530i)17-s + (−5.18 − 3.76i)19-s + (0.309 + 0.951i)20-s + (−0.427 − 3.28i)22-s − 2.59·23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (0.978 − 0.710i)7-s + (−0.286 − 0.207i)8-s + 0.316·10-s + (−0.982 − 0.183i)11-s + (−0.579 − 1.78i)13-s + (0.691 + 0.502i)14-s + (0.0772 − 0.237i)16-s + (−0.0417 + 0.128i)17-s + (−1.18 − 0.864i)19-s + (0.0690 + 0.212i)20-s + (−0.0910 − 0.701i)22-s − 0.541·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.387 + 0.922i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.387 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07026 - 0.711404i\)
\(L(\frac12)\) \(\approx\) \(1.07026 - 0.711404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 \)
5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (3.26 + 0.610i)T \)
good7 \( 1 + (-2.58 + 1.88i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (2.08 + 6.42i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.172 - 0.530i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.18 + 3.76i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.59T + 23T^{2} \)
29 \( 1 + (3.95 - 2.87i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.747 - 2.30i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.63 + 3.36i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.06 - 0.775i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.32T + 43T^{2} \)
47 \( 1 + (4.84 + 3.52i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.752 - 2.31i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.35 + 6.06i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.21 + 3.73i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 + (-1.96 + 6.04i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.61 + 4.08i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.60 - 11.1i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.47 + 7.60i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + (-2.35 - 7.25i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.882067425736759093881096483512, −8.681886294762352338501613738946, −7.976227064416493111764501610573, −7.56639461841754976171033159025, −6.39202556967799535970639489077, −5.23474930200665221685055041673, −4.93234096910549253315056984101, −3.71780919055342677678505444601, −2.34756210822888363145576579856, −0.52154612229098210188136129179, 1.96605085615542638752292212207, 2.36556686463327068523462037296, 3.97216605633858975933279122778, 4.75290224122847335406105012780, 5.68931315905318118110694233814, 6.65321843607801115665325366829, 7.81637194826130582739726091312, 8.515411432619138725747482637016, 9.525501528046303817362346287780, 10.14709447995117903166427368211

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