↑

Properties

Label 2-990-99.32-c1-0-45
Degree $2$
Conductor $990$
Sign $0.813 + 0.581i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (1.55 − 0.767i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (1.44 + 0.961i)6-s + (2.96 − 1.71i)7-s − 0.999·8-s + (1.82 − 2.38i)9-s − 0.999i·10-s + (−1.23 − 3.07i)11-s + (−0.112 + 1.72i)12-s + (−4.74 − 2.73i)13-s + (2.96 + 1.71i)14-s + (−1.72 − 0.112i)15-s + (−0.5 − 0.866i)16-s + 0.378·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.896 − 0.442i)3-s + (−0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.588 + 0.392i)6-s + (1.12 − 0.647i)7-s − 0.353·8-s + (0.607 − 0.794i)9-s − 0.316i·10-s + (−0.372 − 0.927i)11-s + (−0.0323 + 0.498i)12-s + (−1.31 − 0.759i)13-s + (0.792 + 0.457i)14-s + (−0.446 − 0.0289i)15-s + (−0.125 − 0.216i)16-s + 0.0917·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.813 + 0.581i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.813 + 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32324 - 0.744302i\)
\(L(\frac12)\) \(\approx\) \(2.32324 - 0.744302i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-1.55 + 0.767i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (1.23 + 3.07i)T \)
good7 \( 1 + (-2.96 + 1.71i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (4.74 + 2.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.378T + 17T^{2} \)
19 \( 1 + 4.27iT - 19T^{2} \)
23 \( 1 + (-6.38 - 3.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.06 - 3.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.297 + 0.515i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.06T + 37T^{2} \)
41 \( 1 + (3.59 - 6.22i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.81 - 2.77i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.49 + 5.48i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.08iT - 53T^{2} \)
59 \( 1 + (-8.97 - 5.17i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.14 + 4.12i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.73 + 3.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 - 9.64iT - 73T^{2} \)
79 \( 1 + (-12.2 + 7.07i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.713 - 1.23i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.9iT - 89T^{2} \)
97 \( 1 + (-5.56 - 9.63i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.704850531975664999933843188076, −8.630153018791379907312727704905, −8.211398659420167646116572218975, −7.33172328685380948453210692174, −6.96376470572490469019115504430, −5.34324083399191024128921097258, −4.77505474540661662435179767879, −3.57570728204891364768334507091, −2.65622343343785157461955213538, −0.944248073590390485191684212459, 1.92572052595793098398880492839, 2.51074965376168976896075643305, 3.78851326615592731024973015596, 4.77516108640083980518028961252, 5.15318588343549690031084995524, 6.89697251130037133854035545579, 7.70253837048745093555300806345, 8.534772006906482620511287392209, 9.286687109433047531675615459936, 10.17331418059740576638460464410

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