↑

Properties

Label 2-552-24.11-c1-0-43
Degree $2$
Conductor $552$
Sign $0.225 - 0.974i$
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  + (−1.03 + 0.961i)2-s + (1.07 + 1.35i)3-s + (0.152 − 1.99i)4-s + 4.03·5-s + (−2.42 − 0.369i)6-s − 0.282i·7-s + (1.75 + 2.21i)8-s + (−0.672 + 2.92i)9-s + (−4.18 + 3.87i)10-s − 1.95i·11-s + (2.86 − 1.94i)12-s − 0.803i·13-s + (0.271 + 0.293i)14-s + (4.35 + 5.46i)15-s + (−3.95 − 0.609i)16-s + 2.01i·17-s + ⋯
L(s)  = 1  + (−0.733 + 0.679i)2-s + (0.622 + 0.782i)3-s + (0.0763 − 0.997i)4-s + 1.80·5-s + (−0.988 − 0.150i)6-s − 0.106i·7-s + (0.621 + 0.783i)8-s + (−0.224 + 0.974i)9-s + (−1.32 + 1.22i)10-s − 0.588i·11-s + (0.827 − 0.561i)12-s − 0.222i·13-s + (0.0726 + 0.0784i)14-s + (1.12 + 1.41i)15-s + (−0.988 − 0.152i)16-s + 0.489i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.225 - 0.974i$
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: \((2,\ 552,\ (\ :1/2),\ 0.225 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30304 + 1.03556i\)
\(L(\frac12)\) \(\approx\) \(1.30304 + 1.03556i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.03 - 0.961i)T \)
3 \( 1 + (-1.07 - 1.35i)T \)
23 \( 1 + T \)
good5 \( 1 - 4.03T + 5T^{2} \)
7 \( 1 + 0.282iT - 7T^{2} \)
11 \( 1 + 1.95iT - 11T^{2} \)
13 \( 1 + 0.803iT - 13T^{2} \)
17 \( 1 - 2.01iT - 17T^{2} \)
19 \( 1 - 1.56T + 19T^{2} \)
29 \( 1 - 3.72T + 29T^{2} \)
31 \( 1 + 3.80iT - 31T^{2} \)
37 \( 1 + 6.71iT - 37T^{2} \)
41 \( 1 - 7.30iT - 41T^{2} \)
43 \( 1 + 5.37T + 43T^{2} \)
47 \( 1 + 9.50T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 - 9.07iT - 59T^{2} \)
61 \( 1 - 10.4iT - 61T^{2} \)
67 \( 1 - 3.71T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 + 8.93iT - 79T^{2} \)
83 \( 1 + 15.8iT - 83T^{2} \)
89 \( 1 + 2.57iT - 89T^{2} \)
97 \( 1 - 1.99T + 97T^{2} \)
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

−10.31629054446162141468379280412, −10.08870674570948951524459940602, −9.166607256496426823937829452680, −8.617314509637557300873748200922, −7.56285909506083377455717461788, −6.24056782208982006210549249740, −5.66983069620192598563921591782, −4.66058661920423535064080911728, −2.87168439548414446492103157353, −1.65346486984445130454698362363, 1.40592171289513464415831554945, 2.22404560470855128400763812961, 3.18874474324713735650053691707, 4.97023121857515284170715501132, 6.38359939672732149742181536809, 7.00611637261416290025860662572, 8.178328941170599552016724963212, 9.032143913828677909333196731335, 9.690594364957656205269588080289, 10.23183167808399960271809529534

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