↑

Properties

Label 2-552-24.11-c1-0-1
Degree $2$
Conductor $552$
Sign $-0.998 - 0.0509i$
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  + (−0.734 + 1.20i)2-s + (−1.73 + 0.0302i)3-s + (−0.919 − 1.77i)4-s − 0.871·5-s + (1.23 − 2.11i)6-s − 3.09i·7-s + (2.82 + 0.193i)8-s + (2.99 − 0.104i)9-s + (0.640 − 1.05i)10-s + 1.90i·11-s + (1.64 + 3.04i)12-s + 1.23i·13-s + (3.73 + 2.27i)14-s + (1.50 − 0.0263i)15-s + (−2.30 + 3.26i)16-s + 1.98i·17-s + ⋯
L(s)  = 1  + (−0.519 + 0.854i)2-s + (−0.999 + 0.0174i)3-s + (−0.459 − 0.887i)4-s − 0.389·5-s + (0.504 − 0.863i)6-s − 1.16i·7-s + (0.997 + 0.0683i)8-s + (0.999 − 0.0348i)9-s + (0.202 − 0.333i)10-s + 0.573i·11-s + (0.475 + 0.879i)12-s + 0.343i·13-s + (0.998 + 0.607i)14-s + (0.389 − 0.00679i)15-s + (−0.576 + 0.816i)16-s + 0.480i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.998 - 0.0509i$
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: \((2,\ 552,\ (\ :1/2),\ -0.998 - 0.0509i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00520999 + 0.204316i\)
\(L(\frac12)\) \(\approx\) \(0.00520999 + 0.204316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.734 - 1.20i)T \)
3 \( 1 + (1.73 - 0.0302i)T \)
23 \( 1 - T \)
good5 \( 1 + 0.871T + 5T^{2} \)
7 \( 1 + 3.09iT - 7T^{2} \)
11 \( 1 - 1.90iT - 11T^{2} \)
13 \( 1 - 1.23iT - 13T^{2} \)
17 \( 1 - 1.98iT - 17T^{2} \)
19 \( 1 + 4.96T + 19T^{2} \)
29 \( 1 - 2.31T + 29T^{2} \)
31 \( 1 + 3.48iT - 31T^{2} \)
37 \( 1 - 4.56iT - 37T^{2} \)
41 \( 1 - 8.71iT - 41T^{2} \)
43 \( 1 + 3.26T + 43T^{2} \)
47 \( 1 + 7.68T + 47T^{2} \)
53 \( 1 + 7.53T + 53T^{2} \)
59 \( 1 - 12.2iT - 59T^{2} \)
61 \( 1 - 10.7iT - 61T^{2} \)
67 \( 1 + 0.0562T + 67T^{2} \)
71 \( 1 + 7.84T + 71T^{2} \)
73 \( 1 + 7.04T + 73T^{2} \)
79 \( 1 + 5.10iT - 79T^{2} \)
83 \( 1 - 0.795iT - 83T^{2} \)
89 \( 1 - 17.3iT - 89T^{2} \)
97 \( 1 - 9.05T + 97T^{2} \)
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

−10.97871245245473948453930922480, −10.27240392909219561436364932212, −9.655885810292083110393144505864, −8.320319295323061584491174856941, −7.47709740170193321774313894813, −6.74118372596372436067748312101, −6.00425345494566206515680284080, −4.63990285546402189668663057236, −4.15682635631258952590080495368, −1.43063944508947910680365007687, 0.16765982231645026226183185441, 1.97462817917382705109877234058, 3.36993519899815277105935896318, 4.62650621741363915921015839035, 5.60934643989648396725126739616, 6.70880108869148768018368445195, 7.899593498549619905744806636353, 8.713627929686465370964828015636, 9.603205073434371941163512724142, 10.57460479054544051334931917151

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