↑

Properties

Label 2-990-99.32-c1-0-38
Degree $2$
Conductor $990$
Sign $0.999 - 0.0366i$
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 + (0.761 − 1.55i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.5i)5-s + (1.72 − 0.118i)6-s + (2.83 − 1.63i)7-s − 0.999·8-s + (−1.83 − 2.36i)9-s + 0.999i·10-s + (−0.898 + 3.19i)11-s + (0.966 + 1.43i)12-s + (2.49 + 1.44i)13-s + (2.83 + 1.63i)14-s + (1.43 − 0.966i)15-s + (−0.5 − 0.866i)16-s + 1.88·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.439 − 0.898i)3-s + (−0.249 + 0.433i)4-s + (0.387 + 0.223i)5-s + (0.705 − 0.0482i)6-s + (1.06 − 0.617i)7-s − 0.353·8-s + (−0.613 − 0.789i)9-s + 0.316i·10-s + (−0.271 + 0.962i)11-s + (0.278 + 0.414i)12-s + (0.692 + 0.399i)13-s + (0.756 + 0.436i)14-s + (0.371 − 0.249i)15-s + (−0.125 − 0.216i)16-s + 0.456·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.999 - 0.0366i$
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: \((2,\ 990,\ (\ :1/2),\ 0.999 - 0.0366i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59170 + 0.0475078i\)
\(L(\frac12)\) \(\approx\) \(2.59170 + 0.0475078i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.761 + 1.55i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.898 - 3.19i)T \)
good7 \( 1 + (-2.83 + 1.63i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (-2.49 - 1.44i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.88T + 17T^{2} \)
19 \( 1 + 4.33iT - 19T^{2} \)
23 \( 1 + (-6.17 - 3.56i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.688 + 1.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.78 + 3.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.24T + 37T^{2} \)
41 \( 1 + (-3.83 + 6.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.89 + 3.98i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.91 - 5.14i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.25iT - 53T^{2} \)
59 \( 1 + (1.31 + 0.758i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.49 + 3.74i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.84 + 3.20i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 + (11.3 - 6.54i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.462 - 0.801i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.39iT - 89T^{2} \)
97 \( 1 + (8.05 + 13.9i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.791867217317451368747424112012, −8.913408012983683649483489556028, −8.132408068280354743801624837007, −7.19772083211250850301987384764, −6.99001663019477643912873225960, −5.74776717314851652526184901511, −4.86293900572862837408375484795, −3.77057939950788407540236333596, −2.47280966742831850864213155410, −1.29491394821596702913410222425, 1.43053105481580831137207899473, 2.74521024557986642795754367252, 3.55097522712972685879285339342, 4.76837711197937884030632955607, 5.35106232405674403026874935996, 6.13581861773682028923358519081, 7.916668360238112096775173320951, 8.552887084070243445218953360382, 9.097088044932302845031094040626, 10.18139841445639726844811636950

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