Properties

Label 2-24e2-3.2-c2-0-15
Degree $2$
Conductor $576$
Sign $-0.577 + 0.816i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.24i·5-s + 4·7-s − 16.9i·11-s − 8·13-s + 12.7i·17-s − 16·19-s − 16.9i·23-s + 7.00·25-s + 4.24i·29-s − 44·31-s − 16.9i·35-s + 34·37-s − 46.6i·41-s − 40·43-s − 84.8i·47-s + ⋯
L(s)  = 1  − 0.848i·5-s + 0.571·7-s − 1.54i·11-s − 0.615·13-s + 0.748i·17-s − 0.842·19-s − 0.737i·23-s + 0.280·25-s + 0.146i·29-s − 1.41·31-s − 0.484i·35-s + 0.918·37-s − 1.13i·41-s − 0.930·43-s − 1.80i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.284658515\)
\(L(\frac12)\) \(\approx\) \(1.284658515\)
\(L(2)\) 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 \( 1 + 4.24iT - 25T^{2} \)
7 \( 1 - 4T + 49T^{2} \)
11 \( 1 + 16.9iT - 121T^{2} \)
13 \( 1 + 8T + 169T^{2} \)
17 \( 1 - 12.7iT - 289T^{2} \)
19 \( 1 + 16T + 361T^{2} \)
23 \( 1 + 16.9iT - 529T^{2} \)
29 \( 1 - 4.24iT - 841T^{2} \)
31 \( 1 + 44T + 961T^{2} \)
37 \( 1 - 34T + 1.36e3T^{2} \)
41 \( 1 + 46.6iT - 1.68e3T^{2} \)
43 \( 1 + 40T + 1.84e3T^{2} \)
47 \( 1 + 84.8iT - 2.20e3T^{2} \)
53 \( 1 - 38.1iT - 2.80e3T^{2} \)
59 \( 1 + 33.9iT - 3.48e3T^{2} \)
61 \( 1 + 50T + 3.72e3T^{2} \)
67 \( 1 - 8T + 4.48e3T^{2} \)
71 \( 1 + 50.9iT - 5.04e3T^{2} \)
73 \( 1 + 16T + 5.32e3T^{2} \)
79 \( 1 - 76T + 6.24e3T^{2} \)
83 \( 1 + 118. iT - 6.88e3T^{2} \)
89 \( 1 + 12.7iT - 7.92e3T^{2} \)
97 \( 1 - 176T + 9.40e3T^{2} \)
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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

−10.38657295919216017914004676548, −9.017074122597167586942537555400, −8.585294933962246197013940123430, −7.75767836017893870097385507627, −6.46502233235709708214728186314, −5.50745688625015102449690600891, −4.63450192156492300358272123092, −3.48735849955437090272040029836, −1.93885643398718639691324804647, −0.47264380370924817835984330791, 1.80442980206657036402054405532, 2.87543926117971228935150249411, 4.32307144306747344205210136404, 5.13258638695520631463292853335, 6.48972727011012459093414816766, 7.27921972016935093144738772096, 7.920249674073304813048618304025, 9.303229594422882131084680659752, 9.904309879826644061867577309318, 10.87548642493860639251882709288

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