↑

Properties

Label 2-288-72.11-c1-0-1
Degree $2$
Conductor $288$
Sign $-0.249 - 0.968i$
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  + (−0.418 + 1.68i)3-s + (1.60 + 2.78i)5-s + (1.82 + 1.05i)7-s + (−2.64 − 1.40i)9-s + (−3.47 − 2.00i)11-s + (0.341 − 0.197i)13-s + (−5.35 + 1.53i)15-s + 1.20i·17-s + 1.62·19-s + (−2.53 + 2.62i)21-s + (2.74 + 4.75i)23-s + (−2.68 + 4.64i)25-s + (3.47 − 3.86i)27-s + (−2.95 + 5.12i)29-s + (3.34 − 1.93i)31-s + ⋯
L(s)  = 1  + (−0.241 + 0.970i)3-s + (0.719 + 1.24i)5-s + (0.688 + 0.397i)7-s + (−0.883 − 0.469i)9-s + (−1.04 − 0.605i)11-s + (0.0948 − 0.0547i)13-s + (−1.38 + 0.397i)15-s + 0.292i·17-s + 0.372·19-s + (−0.552 + 0.572i)21-s + (0.572 + 0.990i)23-s + (−0.536 + 0.928i)25-s + (0.668 − 0.743i)27-s + (−0.549 + 0.950i)29-s + (0.601 − 0.347i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.249 - 0.968i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.249 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.790104 + 1.01933i\)
\(L(\frac12)\) \(\approx\) \(0.790104 + 1.01933i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.418 - 1.68i)T \)
good5 \( 1 + (-1.60 - 2.78i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.82 - 1.05i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.47 + 2.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.341 + 0.197i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.20iT - 17T^{2} \)
19 \( 1 - 1.62T + 19T^{2} \)
23 \( 1 + (-2.74 - 4.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.95 - 5.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.34 + 1.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 + (1.23 - 0.715i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.21 + 2.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.792 + 1.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.07T + 53T^{2} \)
59 \( 1 + (-2.29 + 1.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.18 - 4.72i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.60 - 4.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.69T + 71T^{2} \)
73 \( 1 - 9.49T + 73T^{2} \)
79 \( 1 + (-1.53 - 0.886i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.30 - 0.755i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.2iT - 89T^{2} \)
97 \( 1 + (-5.84 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.66659761205495136023727146840, −10.93508289760962108467479705992, −10.41277694250757748668369687120, −9.445677170068965062464290248044, −8.396838837344198924134489939348, −7.15104574450181046563577521992, −5.79385989469396222052149893859, −5.27250980926483619434134229528, −3.58286589410824055689446394286, −2.47457251657554182071244831037, 1.08019953706045669843823091059, 2.38835484256368668636953070769, 4.73650071933105811191993220007, 5.34482272729579539112032942149, 6.61195048375203794288277674745, 7.80223194951656415912878382284, 8.436591604652747825560539882529, 9.605237197026576966119786263594, 10.67267124042952811822264342519, 11.75408494774972269066030746124

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