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Properties

Label 2-990-11.3-c1-0-3
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 + (0.309 + 0.951i)8-s + 10-s + (−3.04 + 1.31i)11-s + (1.11 − 0.812i)13-s + (−0.809 − 0.587i)16-s + (1.30 + 0.951i)17-s + (0.618 + 1.90i)19-s + (−0.809 + 0.587i)20-s + (1.69 − 2.85i)22-s − 4.85·23-s + (0.309 + 0.951i)25-s + (−0.427 + 1.31i)26-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.109 + 0.336i)8-s + 0.316·10-s + (−0.918 + 0.396i)11-s + (0.310 − 0.225i)13-s + (−0.202 − 0.146i)16-s + (0.317 + 0.230i)17-s + (0.141 + 0.436i)19-s + (−0.180 + 0.131i)20-s + (0.360 − 0.608i)22-s − 1.01·23-s + (0.0618 + 0.190i)25-s + (−0.0837 + 0.257i)26-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

\(L(1)\) \(\approx\) \(0.248566 + 0.532858i\)
\(L(\frac12)\) \(\approx\) \(0.248566 + 0.532858i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.11 + 0.812i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.30 - 0.951i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.618 - 1.90i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.85T + 23T^{2} \)
29 \( 1 + (2.04 - 6.29i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.35 + 2.43i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.11 - 6.51i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.14 - 3.52i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.85T + 43T^{2} \)
47 \( 1 + (-0.336 - 1.03i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.85 - 6.43i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.354 - 1.08i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.23 - 4.53i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + (-8.85 - 6.43i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.78 - 7.10i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.618 - 0.449i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + (7.47 - 5.42i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.19812579509609528470638791877, −9.464857544857763191748452619192, −8.392569227474798318093346618685, −7.941883468982936159485342396448, −7.11040883743310936853488441121, −6.04306464024351595265988548676, −5.24014446813155831360343069037, −4.19210890511825179073368046418, −2.88034017038644492839041675711, −1.41474288935750994404586462096, 0.33388119557623349571598015757, 2.07859871720755277368258551811, 3.14899514339133260418936038861, 4.12674356362796108548412701978, 5.35519308281434304630887740332, 6.39095102075017989520073234283, 7.42949253753877828506774570240, 8.057562187901403699362768816927, 8.843147606338573136084071673056, 9.815137909250105197124608779818

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