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Properties

Label 2-990-55.13-c1-0-27
Degree $2$
Conductor $990$
Sign $-0.826 + 0.563i$
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  + (0.891 − 0.453i)2-s + (0.587 − 0.809i)4-s + (−0.320 − 2.21i)5-s + (−0.171 − 1.08i)7-s + (0.156 − 0.987i)8-s + (−1.29 − 1.82i)10-s + (−3.30 − 0.231i)11-s + (−1.06 − 2.08i)13-s + (−0.642 − 0.884i)14-s + (−0.309 − 0.951i)16-s + (−1.51 + 2.96i)17-s + (4.05 − 2.94i)19-s + (−1.97 − 1.04i)20-s + (−3.05 + 1.29i)22-s + (−2.67 − 2.67i)23-s + ⋯
L(s)  = 1  + (0.630 − 0.321i)2-s + (0.293 − 0.404i)4-s + (−0.143 − 0.989i)5-s + (−0.0646 − 0.408i)7-s + (0.0553 − 0.349i)8-s + (−0.408 − 0.577i)10-s + (−0.997 − 0.0698i)11-s + (−0.294 − 0.577i)13-s + (−0.171 − 0.236i)14-s + (−0.0772 − 0.237i)16-s + (−0.366 + 0.720i)17-s + (0.930 − 0.675i)19-s + (−0.442 − 0.232i)20-s + (−0.650 + 0.276i)22-s + (−0.557 − 0.557i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.826 + 0.563i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ -0.826 + 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478911 - 1.55355i\)
\(L(\frac12)\) \(\approx\) \(0.478911 - 1.55355i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.891 + 0.453i)T \)
3 \( 1 \)
5 \( 1 + (0.320 + 2.21i)T \)
11 \( 1 + (3.30 + 0.231i)T \)
good7 \( 1 + (0.171 + 1.08i)T + (-6.65 + 2.16i)T^{2} \)
13 \( 1 + (1.06 + 2.08i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (1.51 - 2.96i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (-4.05 + 2.94i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.67 + 2.67i)T + 23iT^{2} \)
29 \( 1 + (1.72 + 1.25i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.615 - 1.89i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.47 + 0.392i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (3.67 + 5.05i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (0.876 - 0.876i)T - 43iT^{2} \)
47 \( 1 + (0.742 - 4.68i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-6.68 + 3.40i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-0.531 + 0.731i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-6.76 + 2.19i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-0.578 + 0.578i)T - 67iT^{2} \)
71 \( 1 + (5.17 + 15.9i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.49 + 0.870i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (1.22 - 3.76i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (10.4 + 5.34i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 - 2.76iT - 89T^{2} \)
97 \( 1 + (6.15 + 12.0i)T + (-57.0 + 78.4i)T^{2} \)
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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

−9.847914938723032073947897427667, −8.806796156921338136852625876069, −7.969213948348410062351561603649, −7.16378235243235111747673988561, −5.90570344882362665542572761564, −5.16976558963553280886410622804, −4.39946652321470249713487405034, −3.35632741814159936964076013988, −2.11135159773949180027454381975, −0.57427921134806407386566540955, 2.20793723875336308562302966189, 3.05982769993293569984344348172, 4.09491635364152528823751168791, 5.27096128780398392743708627125, 5.94467383274417936162175334145, 7.04947888959876760405332094560, 7.50409780538259296249508722655, 8.468657223486403882015076418487, 9.665437526625982558891897070788, 10.28061313939429586031832157046

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