Properties

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

Origins

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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: $\chi_{576} (271, \cdot )$
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 \( 1 + 54.2T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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