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Properties

Label 2-24e2-24.5-c2-0-4
Degree $2$
Conductor $576$
Sign $-0.169 - 0.985i$
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  + 7.34·5-s − 10.3·7-s − 8.48·11-s + 10.3i·13-s + 21.2i·17-s + 20i·19-s + 14.6i·23-s + 29·25-s − 36.7·29-s + 51.9·31-s − 76.3·35-s − 41.5i·37-s + 72.1i·41-s + 40i·43-s − 73.4i·47-s + ⋯
L(s)  = 1  + 1.46·5-s − 1.48·7-s − 0.771·11-s + 0.799i·13-s + 1.24i·17-s + 1.05i·19-s + 0.638i·23-s + 1.15·25-s − 1.26·29-s + 1.67·31-s − 2.18·35-s − 1.12i·37-s + 1.75i·41-s + 0.930i·43-s − 1.56i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.169 - 0.985i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.169 - 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.358897312\)
\(L(\frac12)\) \(\approx\) \(1.358897312\)
\(L(2)\) 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 \( 1 - 7.34T + 25T^{2} \)
7 \( 1 + 10.3T + 49T^{2} \)
11 \( 1 + 8.48T + 121T^{2} \)
13 \( 1 - 10.3iT - 169T^{2} \)
17 \( 1 - 21.2iT - 289T^{2} \)
19 \( 1 - 20iT - 361T^{2} \)
23 \( 1 - 14.6iT - 529T^{2} \)
29 \( 1 + 36.7T + 841T^{2} \)
31 \( 1 - 51.9T + 961T^{2} \)
37 \( 1 + 41.5iT - 1.36e3T^{2} \)
41 \( 1 - 72.1iT - 1.68e3T^{2} \)
43 \( 1 - 40iT - 1.84e3T^{2} \)
47 \( 1 + 73.4iT - 2.20e3T^{2} \)
53 \( 1 - 36.7T + 2.80e3T^{2} \)
59 \( 1 + 33.9T + 3.48e3T^{2} \)
61 \( 1 - 3.72e3T^{2} \)
67 \( 1 - 100iT - 4.48e3T^{2} \)
71 \( 1 - 73.4iT - 5.04e3T^{2} \)
73 \( 1 - 20T + 5.32e3T^{2} \)
79 \( 1 + 51.9T + 6.24e3T^{2} \)
83 \( 1 + 127.T + 6.88e3T^{2} \)
89 \( 1 - 12.7iT - 7.92e3T^{2} \)
97 \( 1 - 40T + 9.40e3T^{2} \)
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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

−10.33744418082488412823514318093, −9.941884308336216910148224041718, −9.293981626717401353536133602286, −8.216252451084953599351018058675, −6.91948087278057519633431953514, −6.08034885894002250511917075248, −5.60269017243032330051084498410, −4.02092084728768168531163057144, −2.80722130209524658017713091541, −1.66396180554149177045964183391, 0.48550264789576945073043845293, 2.44331918331931366998972856504, 3.09137057482780842430955630354, 4.87407676630441023209216187187, 5.75837990759530998414491272686, 6.51346569330790827375436199950, 7.40911863730349474211909162476, 8.809278114244689386349769967554, 9.562987935925959201623382404142, 10.09803315276701945545732587946

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