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Properties

Label 2-990-11.4-c1-0-17
Degree $2$
Conductor $990$
Sign $0.588 + 0.808i$
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 + (−1.07 − 3.31i)7-s + (−0.309 + 0.951i)8-s + 10-s + (2.01 − 2.63i)11-s + (−4.06 − 2.95i)13-s + (1.07 − 3.31i)14-s + (−0.809 + 0.587i)16-s + (−3.05 + 2.21i)17-s + (1.66 − 5.12i)19-s + (0.809 + 0.587i)20-s + (3.17 − 0.951i)22-s + 6.49·23-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.407 − 1.25i)7-s + (−0.109 + 0.336i)8-s + 0.316·10-s + (0.606 − 0.795i)11-s + (−1.12 − 0.818i)13-s + (0.287 − 0.886i)14-s + (−0.202 + 0.146i)16-s + (−0.740 + 0.537i)17-s + (0.382 − 1.17i)19-s + (0.180 + 0.131i)20-s + (0.677 − 0.202i)22-s + 1.35·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.588 + 0.808i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.588 + 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74399 - 0.888114i\)
\(L(\frac12)\) \(\approx\) \(1.74399 - 0.888114i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.01 + 2.63i)T \)
good7 \( 1 + (1.07 + 3.31i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.06 + 2.95i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.05 - 2.21i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.66 + 5.12i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.49T + 23T^{2} \)
29 \( 1 + (0.886 + 2.72i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (7.55 + 5.49i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.73 - 8.42i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.79 + 5.51i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.93T + 43T^{2} \)
47 \( 1 + (3.70 - 11.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.41 - 2.48i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.0224 + 0.0690i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.87 + 6.44i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 15.6T + 67T^{2} \)
71 \( 1 + (3.48 - 2.53i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.85 + 5.70i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.67 - 6.29i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.23 - 0.898i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 3.32T + 89T^{2} \)
97 \( 1 + (5.44 + 3.95i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.745571065195723803491282172262, −9.100511078551917670408231558806, −7.992251539602554943339307095579, −7.16299739624437926547037314955, −6.53213697170820165258419444754, −5.49735168731427314684379218355, −4.61278319988727722088577212973, −3.68692999547767723262629634454, −2.62479162581726710894003238769, −0.73588623193231440831921245305, 1.85647217353799341614999297432, 2.60450242699226702707163333076, 3.78411477983816142990630783076, 4.97514103256098803462797065953, 5.59740216587484532257128778936, 6.74209313546222550578964337886, 7.21671554162040322114963344682, 8.853469428683669596967861654756, 9.367564132957883827530347055624, 9.991262233778940360946038183455

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