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Properties

Label 2-990-33.2-c1-0-10
Degree $2$
Conductor $990$
Sign $-0.170 + 0.985i$
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 + (−0.951 + 0.309i)5-s + (1.97 − 2.71i)7-s + (−0.809 + 0.587i)8-s + 0.999i·10-s + (3.27 + 0.536i)11-s + (2.94 + 0.956i)13-s + (−1.97 − 2.71i)14-s + (0.309 + 0.951i)16-s + (−0.169 − 0.522i)17-s + (−1.98 − 2.72i)19-s + (0.951 + 0.309i)20-s + (1.52 − 2.94i)22-s + 3.45i·23-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.425 + 0.138i)5-s + (0.746 − 1.02i)7-s + (−0.286 + 0.207i)8-s + 0.316i·10-s + (0.986 + 0.161i)11-s + (0.816 + 0.265i)13-s + (−0.527 − 0.726i)14-s + (0.0772 + 0.237i)16-s + (−0.0411 − 0.126i)17-s + (−0.455 − 0.626i)19-s + (0.212 + 0.0690i)20-s + (0.324 − 0.628i)22-s + 0.720i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.170 + 0.985i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.170 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13377 - 1.34736i\)
\(L(\frac12)\) \(\approx\) \(1.13377 - 1.34736i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (-3.27 - 0.536i)T \)
good7 \( 1 + (-1.97 + 2.71i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-2.94 - 0.956i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.169 + 0.522i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.98 + 2.72i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.45iT - 23T^{2} \)
29 \( 1 + (0.129 + 0.0938i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.08 + 9.49i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.95 + 2.87i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.817 + 0.593i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.23iT - 43T^{2} \)
47 \( 1 + (2.87 + 3.96i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.18 + 0.709i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.88 + 8.10i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-8.78 + 2.85i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.23T + 67T^{2} \)
71 \( 1 + (12.4 - 4.05i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.15 - 4.34i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-14.6 - 4.75i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.13 - 6.57i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 1.34iT - 89T^{2} \)
97 \( 1 + (1.71 - 5.26i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.894526093130674942914416295625, −9.009065406707651847995888050182, −8.173277181359395844180747910376, −7.22952360179548957159903340554, −6.40770529694599400307528849103, −5.14432116432073201720773760994, −4.08108753932481588822135542072, −3.72776968130637208225277155257, −2.08523474103960248072240843846, −0.870519964503165642594152598239, 1.46403851872214332642043382972, 3.09186417111089119243183617111, 4.18454007827372135399192900145, 5.04082978092604991039127324206, 6.02322407883396590530715204042, 6.66008804538724782857429108970, 7.85685813988309405198775029686, 8.620780560762141001891488917520, 8.870081881617122039296651380479, 10.18353968510494820146297907099

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