↑

Properties

Label 2-552-24.11-c1-0-47
Degree $2$
Conductor $552$
Sign $0.370 + 0.928i$
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  + (−0.964 − 1.03i)2-s + (−0.513 + 1.65i)3-s + (−0.139 + 1.99i)4-s + 2.51·5-s + (2.20 − 1.06i)6-s − 2.77i·7-s + (2.19 − 1.77i)8-s + (−2.47 − 1.69i)9-s + (−2.42 − 2.59i)10-s − 5.36i·11-s + (−3.22 − 1.25i)12-s + 3.90i·13-s + (−2.86 + 2.67i)14-s + (−1.29 + 4.15i)15-s + (−3.96 − 0.557i)16-s − 6.27i·17-s + ⋯
L(s)  = 1  + (−0.681 − 0.731i)2-s + (−0.296 + 0.955i)3-s + (−0.0698 + 0.997i)4-s + 1.12·5-s + (0.900 − 0.434i)6-s − 1.04i·7-s + (0.777 − 0.629i)8-s + (−0.824 − 0.566i)9-s + (−0.765 − 0.821i)10-s − 1.61i·11-s + (−0.931 − 0.362i)12-s + 1.08i·13-s + (−0.766 + 0.714i)14-s + (−0.333 + 1.07i)15-s + (−0.990 − 0.139i)16-s − 1.52i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.370 + 0.928i$
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: \((2,\ 552,\ (\ :1/2),\ 0.370 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.832426 - 0.564257i\)
\(L(\frac12)\) \(\approx\) \(0.832426 - 0.564257i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.964 + 1.03i)T \)
3 \( 1 + (0.513 - 1.65i)T \)
23 \( 1 - T \)
good5 \( 1 - 2.51T + 5T^{2} \)
7 \( 1 + 2.77iT - 7T^{2} \)
11 \( 1 + 5.36iT - 11T^{2} \)
13 \( 1 - 3.90iT - 13T^{2} \)
17 \( 1 + 6.27iT - 17T^{2} \)
19 \( 1 + 0.842T + 19T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 + 5.16iT - 31T^{2} \)
37 \( 1 + 5.17iT - 37T^{2} \)
41 \( 1 - 8.80iT - 41T^{2} \)
43 \( 1 - 0.848T + 43T^{2} \)
47 \( 1 + 4.84T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 12.0iT - 59T^{2} \)
61 \( 1 - 10.9iT - 61T^{2} \)
67 \( 1 + 1.54T + 67T^{2} \)
71 \( 1 - 1.08T + 71T^{2} \)
73 \( 1 + 1.99T + 73T^{2} \)
79 \( 1 - 9.76iT - 79T^{2} \)
83 \( 1 + 9.45iT - 83T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 - 9.24T + 97T^{2} \)
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

−10.56051138990807013409788969467, −9.742001132603693909665823568721, −9.254073492265256269765919566683, −8.359844689021838311486026170310, −7.00920989157416800604666536609, −6.01622137653269669204485762768, −4.75189186916165696960718947149, −3.73326169717813054053488580749, −2.61251074291813498120517568567, −0.78878816796465204783131613648, 1.58306413046206255601270629956, 2.38903668734645444482914635339, 4.98575939000876394054878143415, 5.74119637268445016520325322877, 6.38601572768576312785407514231, 7.29862113593658189223092393768, 8.308003421318198948080761750827, 8.963429685653727935239759174643, 10.13423600871698776323052621358, 10.50560335469684867202724535232

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