Properties

Label 2-24e2-1.1-c1-0-4
Degree $2$
Conductor $576$
Sign $1$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 4·7-s − 4·11-s + 2·13-s + 6·17-s − 4·19-s − 25-s + 2·29-s − 4·31-s + 8·35-s + 2·37-s − 2·41-s + 4·43-s + 8·47-s + 9·49-s + 10·53-s − 8·55-s + 4·59-s − 6·61-s + 4·65-s + 4·67-s − 16·71-s − 6·73-s − 16·77-s − 4·79-s − 12·83-s + 12·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.51·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 1/5·25-s + 0.371·29-s − 0.718·31-s + 1.35·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s − 1.07·55-s + 0.520·59-s − 0.768·61-s + 0.496·65-s + 0.488·67-s − 1.89·71-s − 0.702·73-s − 1.82·77-s − 0.450·79-s − 1.31·83-s + 1.30·85-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{576} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.912920987\)
\(L(\frac12)\) \(\approx\) \(1.912920987\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−10.59963279276057209057832448649, −10.08615312505774861350749378685, −8.844581178668094717437191348011, −8.093022628739907221893201157530, −7.33135946825865482348621844071, −5.82131382158921014822362291005, −5.37335304530331906156568812203, −4.20875443528614711634884665822, −2.59447091927996366543258857368, −1.47354540780337201919328149138, 1.47354540780337201919328149138, 2.59447091927996366543258857368, 4.20875443528614711634884665822, 5.37335304530331906156568812203, 5.82131382158921014822362291005, 7.33135946825865482348621844071, 8.093022628739907221893201157530, 8.844581178668094717437191348011, 10.08615312505774861350749378685, 10.59963279276057209057832448649

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