↑

Properties

Label 2-288-96.35-c1-0-6
Degree $2$
Conductor $288$
Sign $0.627 + 0.778i$
Analytic cond. $2.29969$
Root an. cond. $1.51647$
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  + (−1.33 − 0.458i)2-s + (1.57 + 1.22i)4-s + (−2.70 + 1.11i)5-s + (−1.06 − 1.06i)7-s + (−1.54 − 2.36i)8-s + (4.12 − 0.257i)10-s + (5.29 − 2.19i)11-s + (1.67 − 4.04i)13-s + (0.937 + 1.91i)14-s + (0.987 + 3.87i)16-s + 3.44·17-s + (3.23 + 1.33i)19-s + (−5.63 − 1.54i)20-s + (−8.08 + 0.505i)22-s + (0.703 + 0.703i)23-s + ⋯
L(s)  = 1  + (−0.945 − 0.324i)2-s + (0.789 + 0.613i)4-s + (−1.20 + 0.500i)5-s + (−0.403 − 0.403i)7-s + (−0.547 − 0.836i)8-s + (1.30 − 0.0814i)10-s + (1.59 − 0.661i)11-s + (0.464 − 1.12i)13-s + (0.250 + 0.512i)14-s + (0.246 + 0.969i)16-s + 0.835·17-s + (0.741 + 0.307i)19-s + (−1.26 − 0.346i)20-s + (−1.72 + 0.107i)22-s + (0.146 + 0.146i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.627 + 0.778i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.627 + 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.639059 - 0.305809i\)
\(L(\frac12)\) \(\approx\) \(0.639059 - 0.305809i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.458i)T \)
3 \( 1 \)
good5 \( 1 + (2.70 - 1.11i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.06 + 1.06i)T + 7iT^{2} \)
11 \( 1 + (-5.29 + 2.19i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.67 + 4.04i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 + (-3.23 - 1.33i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.703 - 0.703i)T + 23iT^{2} \)
29 \( 1 + (-3.94 + 9.52i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 4.23iT - 31T^{2} \)
37 \( 1 + (2.04 + 4.93i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.53 - 3.53i)T - 41iT^{2} \)
43 \( 1 + (3.38 + 8.16i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 4.33iT - 47T^{2} \)
53 \( 1 + (0.541 + 1.30i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.66 - 8.83i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.97 + 0.816i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (3.55 - 8.59i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (1.76 - 1.76i)T - 71iT^{2} \)
73 \( 1 + (1.16 + 1.16i)T + 73iT^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 + (-4.27 + 10.3i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (7.99 + 7.99i)T + 89iT^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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

−11.76954938139496441896889882207, −10.68310172321329211093991723019, −9.915355969060569721211275184122, −8.744510500581509504335171390889, −7.899608928068123263821195951700, −7.08864655973128879525404030354, −6.04551143265209205226743882612, −3.77941097807515366409410650053, −3.27239510296466303446284865527, −0.870702244626014947714807828832, 1.35820215049345926000587681563, 3.48549036921803034665361787420, 4.83707387022176458764009414762, 6.41831553466116078263948010154, 7.12480565346635779355225116514, 8.246874563358345258473594077141, 9.078062279290608677803098446852, 9.688872245167797641578933284119, 11.11194939370438252741542819511, 11.93776391063156349969694392839

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