Properties

Label 2-24e2-1.1-c1-0-6
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·11-s + 2·13-s − 2·17-s − 4·19-s − 8·23-s − 25-s + 6·29-s − 8·31-s − 6·37-s + 6·41-s + 4·43-s − 7·49-s − 2·53-s + 8·55-s − 4·59-s + 2·61-s − 4·65-s − 4·67-s + 8·71-s + 10·73-s + 8·79-s + 4·83-s + 4·85-s + 6·89-s + 8·95-s + 2·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s − 49-s − 0.274·53-s + 1.07·55-s − 0.520·59-s + 0.256·61-s − 0.496·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.900·79-s + 0.439·83-s + 0.433·85-s + 0.635·89-s + 0.820·95-s + 0.203·97-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: \(1\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−10.54464146795679939621303070304, −9.381546175761683500969667585237, −8.240414794588996598510176234846, −7.87582021380809768630528089037, −6.70029239224408692327848930668, −5.67110067949068956351494574607, −4.48990443653583983179988972356, −3.59090248724659668634540115790, −2.18788103243241609268384022128, 0, 2.18788103243241609268384022128, 3.59090248724659668634540115790, 4.48990443653583983179988972356, 5.67110067949068956351494574607, 6.70029239224408692327848930668, 7.87582021380809768630528089037, 8.240414794588996598510176234846, 9.381546175761683500969667585237, 10.54464146795679939621303070304

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