Properties

Label 2-24e2-1.1-c1-0-5
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  + 4·5-s + 6·13-s − 8·17-s + 11·25-s − 4·29-s + 2·37-s + 8·41-s − 7·49-s − 4·53-s + 10·61-s + 24·65-s + 6·73-s − 32·85-s − 16·89-s − 18·97-s − 20·101-s + 6·109-s − 16·113-s + ⋯
L(s)  = 1  + 1.78·5-s + 1.66·13-s − 1.94·17-s + 11/5·25-s − 0.742·29-s + 0.328·37-s + 1.24·41-s − 49-s − 0.549·53-s + 1.28·61-s + 2.97·65-s + 0.702·73-s − 3.47·85-s − 1.69·89-s − 1.82·97-s − 1.99·101-s + 0.574·109-s − 1.50·113-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: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.992332899\)
\(L(\frac12)\) \(\approx\) \(1.992332899\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−10.86823793749660088211696944777, −9.710294823453722661768947413595, −9.107525588245033256035295135392, −8.333949469385387295662212740757, −6.76934202435783070282666272762, −6.20341169361318223940088511214, −5.37375392831683105890386008983, −4.11036595013942989412602319905, −2.57992526666733709076038094728, −1.52256003301293920411212617380, 1.52256003301293920411212617380, 2.57992526666733709076038094728, 4.11036595013942989412602319905, 5.37375392831683105890386008983, 6.20341169361318223940088511214, 6.76934202435783070282666272762, 8.333949469385387295662212740757, 9.107525588245033256035295135392, 9.710294823453722661768947413595, 10.86823793749660088211696944777

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