↑

Properties

Label 2-552-24.11-c1-0-31
Degree $2$
Conductor $552$
Sign $0.973 - 0.227i$
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  + (0.866 − 1.11i)2-s + (−1.39 + 1.02i)3-s + (−0.499 − 1.93i)4-s + 3.31·5-s + (−0.0564 + 2.44i)6-s + 4.93i·7-s + (−2.59 − 1.11i)8-s + (0.882 − 2.86i)9-s + (2.87 − 3.70i)10-s + 2.29i·11-s + (2.68 + 2.18i)12-s + 1.77i·13-s + (5.52 + 4.27i)14-s + (−4.62 + 3.41i)15-s + (−3.50 + 1.93i)16-s + 3.10i·17-s + ⋯
L(s)  = 1  + (0.612 − 0.790i)2-s + (−0.804 + 0.594i)3-s + (−0.249 − 0.968i)4-s + 1.48·5-s + (−0.0230 + 0.999i)6-s + 1.86i·7-s + (−0.918 − 0.395i)8-s + (0.294 − 0.955i)9-s + (0.908 − 1.17i)10-s + 0.693i·11-s + (0.776 + 0.630i)12-s + 0.492i·13-s + (1.47 + 1.14i)14-s + (−1.19 + 0.881i)15-s + (−0.875 + 0.483i)16-s + 0.753i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.973 - 0.227i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.973 - 0.227i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81193 + 0.208647i\)
\(L(\frac12)\) \(\approx\) \(1.81193 + 0.208647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 1.11i)T \)
3 \( 1 + (1.39 - 1.02i)T \)
23 \( 1 - T \)
good5 \( 1 - 3.31T + 5T^{2} \)
7 \( 1 - 4.93iT - 7T^{2} \)
11 \( 1 - 2.29iT - 11T^{2} \)
13 \( 1 - 1.77iT - 13T^{2} \)
17 \( 1 - 3.10iT - 17T^{2} \)
19 \( 1 + 0.712T + 19T^{2} \)
29 \( 1 - 7.87T + 29T^{2} \)
31 \( 1 + 8.99iT - 31T^{2} \)
37 \( 1 + 5.63iT - 37T^{2} \)
41 \( 1 - 4.43iT - 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + 5.16T + 47T^{2} \)
53 \( 1 - 1.07T + 53T^{2} \)
59 \( 1 + 11.0iT - 59T^{2} \)
61 \( 1 + 2.25iT - 61T^{2} \)
67 \( 1 - 6.72T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + 9.31T + 73T^{2} \)
79 \( 1 + 2.14iT - 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 - 4.78iT - 89T^{2} \)
97 \( 1 + 4.06T + 97T^{2} \)
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

−10.90493432580282398654256116041, −9.852019469067235104614528728454, −9.532643635913347014068904928216, −8.729511667606551559549998500093, −6.44536099814060240991421370869, −5.95804626511949613385537551930, −5.29017682054736847979829466246, −4.35729872958759059469997574544, −2.68443554725800604715008691267, −1.80417090357791041192303247520, 1.03755741711648175273465700462, 2.93555450666634199538109372096, 4.49267544308145872299264280559, 5.33337873073573927928010254520, 6.25802626563008516314357843415, 6.88825858121996914203998693712, 7.64818321270423723777565883462, 8.803651968365495957582944239992, 10.18316451895698652762217957920, 10.61014599351068526294971560039

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