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Properties

Label 2-990-33.29-c1-0-5
Degree $2$
Conductor $990$
Sign $0.604 - 0.796i$
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 + (0.587 + 0.809i)5-s + (−1.61 + 0.524i)7-s + (0.309 − 0.951i)8-s − i·10-s + (1.21 + 3.08i)11-s + (3.64 − 5.01i)13-s + (1.61 + 0.524i)14-s + (−0.809 + 0.587i)16-s + (−5.56 + 4.04i)17-s + (2.46 + 0.801i)19-s + (−0.587 + 0.809i)20-s + (0.830 − 3.21i)22-s − 0.389i·23-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (0.262 + 0.361i)5-s + (−0.610 + 0.198i)7-s + (0.109 − 0.336i)8-s − 0.316i·10-s + (0.366 + 0.930i)11-s + (1.01 − 1.39i)13-s + (0.431 + 0.140i)14-s + (−0.202 + 0.146i)16-s + (−1.34 + 0.980i)17-s + (0.565 + 0.183i)19-s + (−0.131 + 0.180i)20-s + (0.177 − 0.684i)22-s − 0.0812i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.604 - 0.796i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.944210 + 0.468514i\)
\(L(\frac12)\) \(\approx\) \(0.944210 + 0.468514i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-1.21 - 3.08i)T \)
good7 \( 1 + (1.61 - 0.524i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.64 + 5.01i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.56 - 4.04i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.46 - 0.801i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.389iT - 23T^{2} \)
29 \( 1 + (0.320 + 0.987i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-6.40 - 4.65i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.965 - 2.97i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.40 - 10.4i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.33iT - 43T^{2} \)
47 \( 1 + (-3.49 - 1.13i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.34 - 3.22i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.21 + 0.395i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.79 + 7.97i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.54T + 67T^{2} \)
71 \( 1 + (-8.36 - 11.5i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.28 - 1.39i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.79 + 6.60i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.87 + 2.08i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 13.5iT - 89T^{2} \)
97 \( 1 + (-14.8 - 10.8i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.09084312455945590614625352295, −9.447341624139664533151240961525, −8.491431537408969207025101657137, −7.82202559741884025923211188222, −6.59041983467993241288657226614, −6.17699487857653927800174459265, −4.74675882480028947011227441962, −3.55336871596919062415048342725, −2.67475838940921816102933274590, −1.34824934863562525812921175844, 0.63660376781456746489672331767, 2.10850486933381914827735301924, 3.58521329037626086233789156804, 4.64214830561486939681020005408, 5.83672226685494862955957548489, 6.55146404156782365213391142784, 7.19172503569679659622730494757, 8.483934950069161459864780086712, 9.035814622361270266680615490450, 9.528739562517880044822354193625

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