Properties

Degree $2$
Conductor $576$
Sign $-0.983 - 0.178i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.909 − 0.909i)5-s + 0.654·7-s + (−13.3 + 13.3i)11-s + (8.32 − 8.32i)13-s + 3.93·17-s + (−16.8 − 16.8i)19-s − 23.1·23-s − 23.3i·25-s + (−35.6 + 35.6i)29-s + 45.5i·31-s + (−0.595 − 0.595i)35-s + (10.1 + 10.1i)37-s − 28.4i·41-s + (−22.7 + 22.7i)43-s − 10.7i·47-s + ⋯
L(s)  = 1  + (−0.181 − 0.181i)5-s + 0.0935·7-s + (−1.21 + 1.21i)11-s + (0.640 − 0.640i)13-s + 0.231·17-s + (−0.889 − 0.889i)19-s − 1.00·23-s − 0.933i·25-s + (−1.22 + 1.22i)29-s + 1.46i·31-s + (−0.0170 − 0.0170i)35-s + (0.274 + 0.274i)37-s − 0.694i·41-s + (−0.528 + 0.528i)43-s − 0.229i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.983 - 0.178i$
Motivic weight: \(2\)
Character: $\chi_{576} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.983 - 0.178i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1337108143\)
\(L(\frac12)\) \(\approx\) \(0.1337108143\)
\(L(2)\) 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 \( 1 + (0.909 + 0.909i)T + 25iT^{2} \)
7 \( 1 - 0.654T + 49T^{2} \)
11 \( 1 + (13.3 - 13.3i)T - 121iT^{2} \)
13 \( 1 + (-8.32 + 8.32i)T - 169iT^{2} \)
17 \( 1 - 3.93T + 289T^{2} \)
19 \( 1 + (16.8 + 16.8i)T + 361iT^{2} \)
23 \( 1 + 23.1T + 529T^{2} \)
29 \( 1 + (35.6 - 35.6i)T - 841iT^{2} \)
31 \( 1 - 45.5iT - 961T^{2} \)
37 \( 1 + (-10.1 - 10.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 28.4iT - 1.68e3T^{2} \)
43 \( 1 + (22.7 - 22.7i)T - 1.84e3iT^{2} \)
47 \( 1 + 10.7iT - 2.20e3T^{2} \)
53 \( 1 + (41.5 + 41.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (21.0 - 21.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (68.7 - 68.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (67.8 + 67.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 33.3T + 5.04e3T^{2} \)
73 \( 1 - 18.6iT - 5.32e3T^{2} \)
79 \( 1 - 6.29iT - 6.24e3T^{2} \)
83 \( 1 + (72.0 + 72.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 10.6iT - 7.92e3T^{2} \)
97 \( 1 - 143.T + 9.40e3T^{2} \)
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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

−10.65635364197221057646778692200, −10.30906300943030757353393332340, −9.123779437065828400629054489748, −8.216203838455472867603617271534, −7.48668438925028028546689931037, −6.44491800419951910812623708053, −5.25994770621548781521330307529, −4.48850853964873346004474524720, −3.13141514325986579992180936933, −1.84095320519560966497208733082, 0.04742396593964375565217998868, 1.92837418465995175804034023601, 3.31124529317290738954087519364, 4.27406342553739340570989192657, 5.71371595906321156719042411844, 6.20814585964559742713092629242, 7.74101630222400779020347449147, 8.105420058255222482336540125505, 9.233589594660006029144533957549, 10.17880805499013993849606933646

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