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Properties

Label 2-990-11.9-c1-0-12
Degree $2$
Conductor $990$
Sign $0.957 - 0.288i$
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 + (−1 − 0.726i)7-s + (0.809 − 0.587i)8-s − 0.999·10-s + (2.54 + 2.12i)11-s + (1.73 − 5.34i)13-s + (1 − 0.726i)14-s + (0.309 + 0.951i)16-s + (−1.57 − 4.84i)17-s + (2.61 − 1.90i)19-s + (0.309 − 0.951i)20-s + (−2.80 + 1.76i)22-s − 0.145·23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (0.138 + 0.425i)5-s + (−0.377 − 0.274i)7-s + (0.286 − 0.207i)8-s − 0.316·10-s + (0.767 + 0.641i)11-s + (0.481 − 1.48i)13-s + (0.267 − 0.194i)14-s + (0.0772 + 0.237i)16-s + (−0.381 − 1.17i)17-s + (0.600 − 0.436i)19-s + (0.0690 − 0.212i)20-s + (−0.598 + 0.375i)22-s − 0.0304·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.957 - 0.288i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.957 - 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36594 + 0.200978i\)
\(L(\frac12)\) \(\approx\) \(1.36594 + 0.200978i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.54 - 2.12i)T \)
good7 \( 1 + (1 + 0.726i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.73 + 5.34i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.57 + 4.84i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.61 + 1.90i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.145T + 23T^{2} \)
29 \( 1 + (0.309 + 0.224i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.11 - 3.44i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.97 - 5.06i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.61 - 1.90i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + (-9.16 + 6.65i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.145 + 0.449i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.35 - 1.71i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 6.09T + 67T^{2} \)
71 \( 1 + (0.472 + 1.45i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-10.8 - 7.88i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.19 + 6.74i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.23 + 3.80i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + (3.90 - 12.0i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.847707960625099424416522299918, −9.272860417706930132150137998519, −8.277988308542370758231817386539, −7.32313594860022018157515845306, −6.81691883108937638695629785148, −5.84501038484830960618476921468, −4.96205060871847746808895466745, −3.79173904091858630169353354345, −2.68558127962553740927858939752, −0.852414402179052505549438273052, 1.18019166059743753499749888317, 2.30646821536865223894530627712, 3.75811312048361660795530212755, 4.27382036125773552504466382648, 5.76323523845476693930623051191, 6.37485891693583442620951007950, 7.60073571298115514089191549623, 8.653719425222854379069224688625, 9.160634919797473881647052032385, 9.764284514803159398506523006516

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