Properties

Degree $2$
Conductor $576$
Sign $0.893 - 0.449i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.24 + 5.24i)5-s + 5.32·7-s + (12.2 − 12.2i)11-s + (−5.73 + 5.73i)13-s + 23.3·17-s + (−11.7 − 11.7i)19-s + 5.80·23-s + 29.9i·25-s + (−18.3 + 18.3i)29-s − 16.9i·31-s + (27.9 + 27.9i)35-s + (15.3 + 15.3i)37-s − 29.2i·41-s + (−33.4 + 33.4i)43-s + 18.2i·47-s + ⋯
L(s)  = 1  + (1.04 + 1.04i)5-s + 0.761·7-s + (1.11 − 1.11i)11-s + (−0.441 + 0.441i)13-s + 1.37·17-s + (−0.618 − 0.618i)19-s + 0.252·23-s + 1.19i·25-s + (−0.634 + 0.634i)29-s − 0.545i·31-s + (0.798 + 0.798i)35-s + (0.414 + 0.414i)37-s − 0.713i·41-s + (−0.776 + 0.776i)43-s + 0.387i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.893 - 0.449i$
Motivic weight: \(2\)
Character: $\chi_{576} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.893 - 0.449i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.521108622\)
\(L(\frac12)\) \(\approx\) \(2.521108622\)
\(L(2)\) 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 \( 1 + (-5.24 - 5.24i)T + 25iT^{2} \)
7 \( 1 - 5.32T + 49T^{2} \)
11 \( 1 + (-12.2 + 12.2i)T - 121iT^{2} \)
13 \( 1 + (5.73 - 5.73i)T - 169iT^{2} \)
17 \( 1 - 23.3T + 289T^{2} \)
19 \( 1 + (11.7 + 11.7i)T + 361iT^{2} \)
23 \( 1 - 5.80T + 529T^{2} \)
29 \( 1 + (18.3 - 18.3i)T - 841iT^{2} \)
31 \( 1 + 16.9iT - 961T^{2} \)
37 \( 1 + (-15.3 - 15.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 29.2iT - 1.68e3T^{2} \)
43 \( 1 + (33.4 - 33.4i)T - 1.84e3iT^{2} \)
47 \( 1 - 18.2iT - 2.20e3T^{2} \)
53 \( 1 + (-66.9 - 66.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (27.1 - 27.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (-65.2 + 65.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (-37.6 - 37.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 42.6T + 5.04e3T^{2} \)
73 \( 1 + 106. iT - 5.32e3T^{2} \)
79 \( 1 + 21.2iT - 6.24e3T^{2} \)
83 \( 1 + (-24.1 - 24.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 52.8iT - 7.92e3T^{2} \)
97 \( 1 + 21.0T + 9.40e3T^{2} \)
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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

−10.67620289586406752740744171905, −9.701463214492395145063842912709, −8.986800413878124628833007833081, −7.88020895197566074396370643840, −6.81648557190474265421272948243, −6.11509110377582135670470602776, −5.17470996959424490623104621077, −3.74839930693501885865416675391, −2.61045428284715148001340726264, −1.35680123514872869661549488509, 1.22021903795105901477657515676, 2.07169258061870525796701193606, 3.92843697519877843091533639427, 5.00443690713875292027961537606, 5.62546739370187128104741457610, 6.81882771212165910798105164258, 7.925103110549683078590225450370, 8.742472483295503297634875923751, 9.755652665762294795933252643754, 10.04976109197476677090116839709

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