Properties

Label 2-552-184.99-c1-0-34
Degree $2$
Conductor $552$
Sign $0.547 + 0.836i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
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  + (−0.963 + 1.03i)2-s + (0.841 − 0.540i)3-s + (−0.141 − 1.99i)4-s + (−0.831 − 1.81i)5-s + (−0.251 + 1.39i)6-s + (2.23 − 0.657i)7-s + (2.20 + 1.77i)8-s + (0.415 − 0.909i)9-s + (2.68 + 0.894i)10-s + (−2.04 + 1.77i)11-s + (−1.19 − 1.60i)12-s + (0.981 − 3.34i)13-s + (−1.47 + 2.94i)14-s + (−1.68 − 1.08i)15-s + (−3.95 + 0.564i)16-s + (1.42 + 0.205i)17-s + ⋯
L(s)  = 1  + (−0.681 + 0.731i)2-s + (0.485 − 0.312i)3-s + (−0.0707 − 0.997i)4-s + (−0.371 − 0.813i)5-s + (−0.102 + 0.568i)6-s + (0.846 − 0.248i)7-s + (0.778 + 0.628i)8-s + (0.138 − 0.303i)9-s + (0.848 + 0.282i)10-s + (−0.616 + 0.533i)11-s + (−0.345 − 0.462i)12-s + (0.272 − 0.927i)13-s + (−0.394 + 0.788i)14-s + (−0.434 − 0.279i)15-s + (−0.989 + 0.141i)16-s + (0.345 + 0.0497i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.547 + 0.836i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.547 + 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.965160 - 0.521971i\)
\(L(\frac12)\) \(\approx\) \(0.965160 - 0.521971i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.963 - 1.03i)T \)
3 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (-0.00837 + 4.79i)T \)
good5 \( 1 + (0.831 + 1.81i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (-2.23 + 0.657i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (2.04 - 1.77i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (-0.981 + 3.34i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (-1.42 - 0.205i)T + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (2.66 - 0.382i)T + (18.2 - 5.35i)T^{2} \)
29 \( 1 + (5.85 + 0.842i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (-3.77 + 5.87i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 + (-2.39 + 5.23i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-0.234 - 0.514i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-1.51 - 2.35i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 - 0.481iT - 47T^{2} \)
53 \( 1 + (2.44 - 0.717i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (0.652 + 0.191i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-0.775 - 0.498i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (-7.44 - 6.45i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (-11.3 - 9.86i)T + (10.1 + 70.2i)T^{2} \)
73 \( 1 + (2.05 + 14.2i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-2.89 - 0.848i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-2.80 - 1.28i)T + (54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.0629 - 0.0980i)T + (-36.9 + 80.9i)T^{2} \)
97 \( 1 + (0.220 - 0.100i)T + (63.5 - 73.3i)T^{2} \)
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

−10.48153132431569217695229232065, −9.565310394124300286806149081122, −8.536060880920197272311774177589, −8.017462814020109641886138758769, −7.47805704399117697125678698551, −6.17675131514781650845071499001, −5.10272079984186577803618755065, −4.23485125486204946992041652140, −2.23286173829351813289482727606, −0.78096332624045207858724255803, 1.76363640236325240671206465257, 2.97119862043508636941719052060, 3.85792441988476239746035787263, 5.07891015530281454002347141224, 6.72201318759789194096601834560, 7.69536432982142779643376700695, 8.354444126413793998975896091842, 9.180584722363365710203183108287, 10.11167589118973165364694704219, 11.13226071394515901011917006838

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