↑

Properties

Label 2-288-32.5-c1-0-1
Degree $2$
Conductor $288$
Sign $-0.417 - 0.908i$
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  + (−0.884 − 1.10i)2-s + (−0.436 + 1.95i)4-s + (−2.14 − 0.890i)5-s + (−1.10 − 1.10i)7-s + (2.54 − 1.24i)8-s + (0.917 + 3.16i)10-s + (−0.999 + 2.41i)11-s + (−2.03 + 0.841i)13-s + (−0.241 + 2.18i)14-s + (−3.61 − 1.70i)16-s + 5.68i·17-s + (−6.02 + 2.49i)19-s + (2.67 − 3.80i)20-s + (3.54 − 1.02i)22-s + (−3.60 + 3.60i)23-s + ⋯
L(s)  = 1  + (−0.625 − 0.780i)2-s + (−0.218 + 0.975i)4-s + (−0.961 − 0.398i)5-s + (−0.415 − 0.415i)7-s + (0.898 − 0.439i)8-s + (0.290 + 0.999i)10-s + (−0.301 + 0.727i)11-s + (−0.563 + 0.233i)13-s + (−0.0646 + 0.584i)14-s + (−0.904 − 0.426i)16-s + 1.37i·17-s + (−1.38 + 0.572i)19-s + (0.598 − 0.851i)20-s + (0.756 − 0.219i)22-s + (−0.751 + 0.751i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.417 - 0.908i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.417 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0577438 + 0.0901202i\)
\(L(\frac12)\) \(\approx\) \(0.0577438 + 0.0901202i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.884 + 1.10i)T \)
3 \( 1 \)
good5 \( 1 + (2.14 + 0.890i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.10 + 1.10i)T + 7iT^{2} \)
11 \( 1 + (0.999 - 2.41i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.03 - 0.841i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 5.68iT - 17T^{2} \)
19 \( 1 + (6.02 - 2.49i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (3.60 - 3.60i)T - 23iT^{2} \)
29 \( 1 + (3.82 + 9.23i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.98T + 31T^{2} \)
37 \( 1 + (-5.97 - 2.47i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-4.33 + 4.33i)T - 41iT^{2} \)
43 \( 1 + (4.39 - 10.6i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 5.32iT - 47T^{2} \)
53 \( 1 + (0.802 - 1.93i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (5.97 + 2.47i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (3.53 + 8.54i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (2.25 + 5.44i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (2.57 + 2.57i)T + 71iT^{2} \)
73 \( 1 + (-8.01 + 8.01i)T - 73iT^{2} \)
79 \( 1 - 14.4iT - 79T^{2} \)
83 \( 1 + (2.50 - 1.03i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-2.98 - 2.98i)T + 89iT^{2} \)
97 \( 1 + 5.81T + 97T^{2} \)
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

−12.11146963949257159489850819192, −11.15720356709909467106184775893, −10.18183384285669637273086823504, −9.504331035170666855288681250363, −8.085479845257299672814513950242, −7.83250027903138147438180579418, −6.39424416876414748545034947634, −4.41624158775175310798612954411, −3.81020213209407515857759572760, −2.04259380623980369775865402930, 0.093282932653010814885919695039, 2.76218205068535942129288055645, 4.44307741209642433768701366342, 5.68711356151244246431362816317, 6.80457738164716658142062859890, 7.60188331102326027659403850373, 8.562488239171080254412480494010, 9.390287102108387375969385433679, 10.57367005108530665960422068676, 11.25459117139493231784820555117

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