Properties

Degree $2$
Conductor $576$
Sign $-0.495 - 0.868i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.32 + 3.32i)5-s + 4.04·7-s + (6.82 + 6.82i)11-s + (4.29 + 4.29i)13-s − 30.1·17-s + (19.7 − 19.7i)19-s − 28.2·23-s + 2.86i·25-s + (21.3 + 21.3i)29-s + 38.0i·31-s + (−13.4 + 13.4i)35-s + (−42.8 + 42.8i)37-s + 48.2i·41-s + (−32.6 − 32.6i)43-s + 15.8i·47-s + ⋯
L(s)  = 1  + (−0.665 + 0.665i)5-s + 0.577·7-s + (0.620 + 0.620i)11-s + (0.330 + 0.330i)13-s − 1.77·17-s + (1.03 − 1.03i)19-s − 1.22·23-s + 0.114i·25-s + (0.736 + 0.736i)29-s + 1.22i·31-s + (−0.384 + 0.384i)35-s + (−1.15 + 1.15i)37-s + 1.17i·41-s + (−0.759 − 0.759i)43-s + 0.336i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.495 - 0.868i$
Motivic weight: \(2\)
Character: $\chi_{576} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.495 - 0.868i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.147559076\)
\(L(\frac12)\) \(\approx\) \(1.147559076\)
\(L(2)\) 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 \( 1 + (3.32 - 3.32i)T - 25iT^{2} \)
7 \( 1 - 4.04T + 49T^{2} \)
11 \( 1 + (-6.82 - 6.82i)T + 121iT^{2} \)
13 \( 1 + (-4.29 - 4.29i)T + 169iT^{2} \)
17 \( 1 + 30.1T + 289T^{2} \)
19 \( 1 + (-19.7 + 19.7i)T - 361iT^{2} \)
23 \( 1 + 28.2T + 529T^{2} \)
29 \( 1 + (-21.3 - 21.3i)T + 841iT^{2} \)
31 \( 1 - 38.0iT - 961T^{2} \)
37 \( 1 + (42.8 - 42.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 48.2iT - 1.68e3T^{2} \)
43 \( 1 + (32.6 + 32.6i)T + 1.84e3iT^{2} \)
47 \( 1 - 15.8iT - 2.20e3T^{2} \)
53 \( 1 + (-0.476 + 0.476i)T - 2.80e3iT^{2} \)
59 \( 1 + (-9.97 - 9.97i)T + 3.48e3iT^{2} \)
61 \( 1 + (-37.9 - 37.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (20.0 - 20.0i)T - 4.48e3iT^{2} \)
71 \( 1 - 40.0T + 5.04e3T^{2} \)
73 \( 1 + 30.8iT - 5.32e3T^{2} \)
79 \( 1 - 130. iT - 6.24e3T^{2} \)
83 \( 1 + (2.26 - 2.26i)T - 6.88e3iT^{2} \)
89 \( 1 + 72.2iT - 7.92e3T^{2} \)
97 \( 1 + 112.T + 9.40e3T^{2} \)
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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

−11.01144307648749288963317064936, −9.999423613640775922572961008045, −8.951340646114726509853461430777, −8.210477926044699882533305522997, −6.96084199088689727098817524120, −6.71219761955243335780309239962, −5.05211356799268169645769110820, −4.22224998538855778180401457383, −3.05410649004038496248537465021, −1.62591355348961896668118485425, 0.44056685559533915395093941672, 1.94914570498028223489105587364, 3.68703622782947110549252860320, 4.41266494029972670681322536902, 5.58013151926988777100419956232, 6.54771817160936054394816213159, 7.84657685768290148028509034711, 8.353696162618528676442410939773, 9.187430972841184089161102020815, 10.27033084785308415548407749082

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