↑

Properties

Label 2-288-32.13-c1-0-12
Degree $2$
Conductor $288$
Sign $0.800 - 0.599i$
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  + (1.26 + 0.628i)2-s + (1.20 + 1.59i)4-s + (3.09 − 1.28i)5-s + (−1.73 + 1.73i)7-s + (0.531 + 2.77i)8-s + (4.72 + 0.321i)10-s + (−2.39 − 5.79i)11-s + (0.0173 + 0.00717i)13-s + (−3.28 + 1.10i)14-s + (−1.07 + 3.85i)16-s + 5.57i·17-s + (−1.03 − 0.426i)19-s + (5.78 + 3.37i)20-s + (0.601 − 8.84i)22-s + (−2.01 − 2.01i)23-s + ⋯
L(s)  = 1  + (0.895 + 0.444i)2-s + (0.604 + 0.796i)4-s + (1.38 − 0.572i)5-s + (−0.655 + 0.655i)7-s + (0.187 + 0.982i)8-s + (1.49 + 0.101i)10-s + (−0.723 − 1.74i)11-s + (0.00480 + 0.00199i)13-s + (−0.878 + 0.295i)14-s + (−0.268 + 0.963i)16-s + 1.35i·17-s + (−0.236 − 0.0979i)19-s + (1.29 + 0.754i)20-s + (0.128 − 1.88i)22-s + (−0.420 − 0.420i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.800 - 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20873 + 0.734830i\)
\(L(\frac12)\) \(\approx\) \(2.20873 + 0.734830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.628i)T \)
3 \( 1 \)
good5 \( 1 + (-3.09 + 1.28i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.73 - 1.73i)T - 7iT^{2} \)
11 \( 1 + (2.39 + 5.79i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.0173 - 0.00717i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 5.57iT - 17T^{2} \)
19 \( 1 + (1.03 + 0.426i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.01 + 2.01i)T + 23iT^{2} \)
29 \( 1 + (-0.706 + 1.70i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.38T + 31T^{2} \)
37 \( 1 + (2.87 - 1.19i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (6.97 + 6.97i)T + 41iT^{2} \)
43 \( 1 + (-1.67 - 4.03i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 1.15iT - 47T^{2} \)
53 \( 1 + (2.56 + 6.19i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.735 + 0.304i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-4.82 + 11.6i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (2.05 - 4.97i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-1.78 + 1.78i)T - 71iT^{2} \)
73 \( 1 + (-1.67 - 1.67i)T + 73iT^{2} \)
79 \( 1 + 2.67iT - 79T^{2} \)
83 \( 1 + (-6.91 - 2.86i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (6.73 - 6.73i)T - 89iT^{2} \)
97 \( 1 + 1.75T + 97T^{2} \)
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

−12.32145564289994219819891352669, −11.04336753575841744590320574840, −10.09421273357285590023637204282, −8.790277358883067300539182663132, −8.226417180527996763482592900510, −6.38746661954037261304532084505, −5.94194257965732806516614194341, −5.13953238573148623180902303071, −3.46755853621790831170417826110, −2.22113103198163043638138418763, 1.95718253799337200373923012475, 2.99933570799866242859530387102, 4.56715257152792688946146191837, 5.56154446710790566015815506226, 6.72032245626384017060004119048, 7.28800674333394298643410721683, 9.515354368620435834020684994062, 10.00980975769779157652710062169, 10.57994809019537993282102222535, 11.88286609810088562714831155550

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