↑

Properties

Label 2-2268-21.20-c3-0-42
Degree $2$
Conductor $2268$
Sign $0.764 - 0.644i$
Analytic cond. $133.816$
Root an. cond. $11.5679$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 16.5·5-s + (−11.9 − 14.1i)7-s + 53.4i·11-s + 12.7i·13-s − 96.7·17-s − 54.6i·19-s + 64.2i·23-s + 150.·25-s − 130. i·29-s − 220. i·31-s + (−198. − 234. i)35-s + 279.·37-s + 370.·41-s + 307.·43-s − 327.·47-s + ⋯
L(s)  = 1  + 1.48·5-s + (−0.644 − 0.764i)7-s + 1.46i·11-s + 0.272i·13-s − 1.38·17-s − 0.660i·19-s + 0.582i·23-s + 1.20·25-s − 0.833i·29-s − 1.27i·31-s + (−0.957 − 1.13i)35-s + 1.24·37-s + 1.41·41-s + 1.09·43-s − 1.01·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(133.816\)
Root analytic conductor: \(11.5679\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :3/2),\ 0.764 - 0.644i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.459340263\)
\(L(\frac12)\) \(\approx\) \(2.459340263\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (11.9 + 14.1i)T \)
good5 \( 1 - 16.5T + 125T^{2} \)
11 \( 1 - 53.4iT - 1.33e3T^{2} \)
13 \( 1 - 12.7iT - 2.19e3T^{2} \)
17 \( 1 + 96.7T + 4.91e3T^{2} \)
19 \( 1 + 54.6iT - 6.85e3T^{2} \)
23 \( 1 - 64.2iT - 1.21e4T^{2} \)
29 \( 1 + 130. iT - 2.43e4T^{2} \)
31 \( 1 + 220. iT - 2.97e4T^{2} \)
37 \( 1 - 279.T + 5.06e4T^{2} \)
41 \( 1 - 370.T + 6.89e4T^{2} \)
43 \( 1 - 307.T + 7.95e4T^{2} \)
47 \( 1 + 327.T + 1.03e5T^{2} \)
53 \( 1 - 451. iT - 1.48e5T^{2} \)
59 \( 1 + 517.T + 2.05e5T^{2} \)
61 \( 1 - 270. iT - 2.26e5T^{2} \)
67 \( 1 - 741.T + 3.00e5T^{2} \)
71 \( 1 - 914. iT - 3.57e5T^{2} \)
73 \( 1 - 337. iT - 3.89e5T^{2} \)
79 \( 1 - 997.T + 4.93e5T^{2} \)
83 \( 1 + 34.0T + 5.71e5T^{2} \)
89 \( 1 + 208.T + 7.04e5T^{2} \)
97 \( 1 - 1.27e3iT - 9.12e5T^{2} \)
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

−9.294609820879210887106088453640, −7.81300248985850886169787264570, −7.09262427775124315744848383510, −6.40270805401107240988202437746, −5.80832403903324686399377709285, −4.61201826738446605626529148587, −4.14328187776743018406640504426, −2.57367258170363855533740806349, −2.10536717345081819774075898564, −0.881449576501817120866296330813, 0.55064687047889212799302337750, 1.82670937396879043011376896789, 2.66540911520666829874380702056, 3.41910436568934455194020868424, 4.79664337097584384908197427018, 5.64488344286343939643208929484, 6.22184158972482292900698490927, 6.62038018350126953655886044353, 8.004573495256868090710035473659, 8.852605235727104978007355757659

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