↑

Properties

Label 2-288-24.5-c2-0-7
Degree $2$
Conductor $288$
Sign $-0.888 + 0.458i$
Analytic cond. $7.84743$
Root an. cond. $2.80132$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.07·5-s − 7.21·7-s − 16.3·11-s − 21.6i·13-s + 18.9i·17-s − 17.0i·19-s − 1.11i·23-s − 23.8·25-s − 29.4·29-s − 5.63·31-s − 7.75·35-s − 17.0i·37-s − 27.4i·41-s − 52.3i·43-s + 64.5i·47-s + ⋯
L(s)  = 1  + 0.214·5-s − 1.03·7-s − 1.48·11-s − 1.66i·13-s + 1.11i·17-s − 0.897i·19-s − 0.0485i·23-s − 0.953·25-s − 1.01·29-s − 0.181·31-s − 0.221·35-s − 0.460i·37-s − 0.669i·41-s − 1.21i·43-s + 1.37i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.888 + 0.458i$
Analytic conductor: \(7.84743\)
Root analytic conductor: \(2.80132\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ -0.888 + 0.458i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0946009 - 0.389401i\)
\(L(\frac12)\) \(\approx\) \(0.0946009 - 0.389401i\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 - 1.07T + 25T^{2} \)
7 \( 1 + 7.21T + 49T^{2} \)
11 \( 1 + 16.3T + 121T^{2} \)
13 \( 1 + 21.6iT - 169T^{2} \)
17 \( 1 - 18.9iT - 289T^{2} \)
19 \( 1 + 17.0iT - 361T^{2} \)
23 \( 1 + 1.11iT - 529T^{2} \)
29 \( 1 + 29.4T + 841T^{2} \)
31 \( 1 + 5.63T + 961T^{2} \)
37 \( 1 + 17.0iT - 1.36e3T^{2} \)
41 \( 1 + 27.4iT - 1.68e3T^{2} \)
43 \( 1 + 52.3iT - 1.84e3T^{2} \)
47 \( 1 - 64.5iT - 2.20e3T^{2} \)
53 \( 1 + 35.9T + 2.80e3T^{2} \)
59 \( 1 - 56.8T + 3.48e3T^{2} \)
61 \( 1 - 69.3iT - 3.72e3T^{2} \)
67 \( 1 - 69.3iT - 4.48e3T^{2} \)
71 \( 1 + 98.4iT - 5.04e3T^{2} \)
73 \( 1 - 37.6T + 5.32e3T^{2} \)
79 \( 1 - 127.T + 6.24e3T^{2} \)
83 \( 1 - 7.75T + 6.88e3T^{2} \)
89 \( 1 - 76.1iT - 7.92e3T^{2} \)
97 \( 1 - 4.84T + 9.40e3T^{2} \)
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

−10.87878225975153560143173254277, −10.35337767807109435365363945882, −9.428148420590457730934923085224, −8.213471619275300623517891865703, −7.38357572850283577420618815943, −6.02132869472385623421615496228, −5.30605546575634565808190620726, −3.61016671107641941116586807777, −2.49525561757794854702562045339, −0.18111100150737372037929251956, 2.16769381196155877109088458642, 3.51624532274301996575875238235, 4.93171713760566414678758864516, 6.08296970750250436553094013455, 7.06784105055296079172561940095, 8.100054371167048167043873526003, 9.476379656365848315733757327545, 9.833755907907231659025565910146, 11.08914673882422820099015576380, 11.97133577408137245769834081325

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