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Properties

Label 2-24e2-1.1-c1-0-8
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  − 4·7-s − 2·13-s − 8·19-s − 5·25-s − 4·31-s + 10·37-s − 8·43-s + 9·49-s − 14·61-s + 16·67-s − 10·73-s − 4·79-s + 8·91-s + 14·97-s + 20·103-s − 2·109-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.554·13-s − 1.83·19-s − 25-s − 0.718·31-s + 1.64·37-s − 1.21·43-s + 9/7·49-s − 1.79·61-s + 1.95·67-s − 1.17·73-s − 0.450·79-s + 0.838·91-s + 1.42·97-s + 1.97·103-s − 0.191·109-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: Trivial
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 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−10.12697942185622269661894542118, −9.549312103602254150736123536576, −8.627688042848610733959539861310, −7.53350611206671162396542910213, −6.53610172162150019204913040571, −5.92154790799829788010807237440, −4.50007207375409781750664997050, −3.45625920459556711940880113888, −2.26336509042736485238326670477, 0, 2.26336509042736485238326670477, 3.45625920459556711940880113888, 4.50007207375409781750664997050, 5.92154790799829788010807237440, 6.53610172162150019204913040571, 7.53350611206671162396542910213, 8.627688042848610733959539861310, 9.549312103602254150736123536576, 10.12697942185622269661894542118

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