Properties

Degree $2$
Conductor $576$
Sign $-0.479 + 0.877i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−6.49 + 6.49i)5-s − 3.94·7-s + (4.31 + 4.31i)11-s + (4.06 + 4.06i)13-s + 14.5·17-s + (−4.94 + 4.94i)19-s − 43.6·23-s − 59.3i·25-s + (−25.0 − 25.0i)29-s − 32.5i·31-s + (25.6 − 25.6i)35-s + (4.14 − 4.14i)37-s − 55.3i·41-s + (16.1 + 16.1i)43-s − 7.92i·47-s + ⋯
L(s)  = 1  + (−1.29 + 1.29i)5-s − 0.563·7-s + (0.391 + 0.391i)11-s + (0.312 + 0.312i)13-s + 0.856·17-s + (−0.260 + 0.260i)19-s − 1.89·23-s − 2.37i·25-s + (−0.865 − 0.865i)29-s − 1.04i·31-s + (0.731 − 0.731i)35-s + (0.111 − 0.111i)37-s − 1.34i·41-s + (0.374 + 0.374i)43-s − 0.168i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.479 + 0.877i$
Motivic weight: \(2\)
Character: $\chi_{576} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.479 + 0.877i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1334756923\)
\(L(\frac12)\) \(\approx\) \(0.1334756923\)
\(L(2)\) 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 \( 1 + (6.49 - 6.49i)T - 25iT^{2} \)
7 \( 1 + 3.94T + 49T^{2} \)
11 \( 1 + (-4.31 - 4.31i)T + 121iT^{2} \)
13 \( 1 + (-4.06 - 4.06i)T + 169iT^{2} \)
17 \( 1 - 14.5T + 289T^{2} \)
19 \( 1 + (4.94 - 4.94i)T - 361iT^{2} \)
23 \( 1 + 43.6T + 529T^{2} \)
29 \( 1 + (25.0 + 25.0i)T + 841iT^{2} \)
31 \( 1 + 32.5iT - 961T^{2} \)
37 \( 1 + (-4.14 + 4.14i)T - 1.36e3iT^{2} \)
41 \( 1 + 55.3iT - 1.68e3T^{2} \)
43 \( 1 + (-16.1 - 16.1i)T + 1.84e3iT^{2} \)
47 \( 1 + 7.92iT - 2.20e3T^{2} \)
53 \( 1 + (-31.5 + 31.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (49.7 + 49.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (-44.4 - 44.4i)T + 3.72e3iT^{2} \)
67 \( 1 + (-1.64 + 1.64i)T - 4.48e3iT^{2} \)
71 \( 1 - 24.1T + 5.04e3T^{2} \)
73 \( 1 - 10.7iT - 5.32e3T^{2} \)
79 \( 1 - 72.0iT - 6.24e3T^{2} \)
83 \( 1 + (-42.0 + 42.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 28.9iT - 7.92e3T^{2} \)
97 \( 1 + 54.2T + 9.40e3T^{2} \)
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

−10.24516824032723540376921201598, −9.568705439684257579662263678310, −8.161850891132024034269378576589, −7.58332905375894064961409497141, −6.67746588861462985969081745275, −5.88508371514830018261760441695, −4.04545674680468532054991224370, −3.67606786340911939475609061560, −2.30670055415851866986010707328, −0.05558082001316149636806432381, 1.25302551976345386718911768976, 3.34216858167658467942540071708, 4.08544549612399562657140372244, 5.16540519648656478151198121711, 6.19460271592769192853517934745, 7.48190714042170439130695913414, 8.205145644882893361463695659130, 8.903956428429186173044359921004, 9.803393409938735206622369770496, 10.93241272464632035274184840864

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