Properties

Label 2-24e2-1.1-c1-0-7
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 − 6·13-s − 2·17-s − 25-s − 10·29-s + 2·37-s − 10·41-s − 7·49-s + 14·53-s + 10·61-s + 12·65-s − 6·73-s + 4·85-s − 10·89-s + 18·97-s − 2·101-s − 6·109-s + 14·113-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.66·13-s − 0.485·17-s − 1/5·25-s − 1.85·29-s + 0.328·37-s − 1.56·41-s − 49-s + 1.92·53-s + 1.28·61-s + 1.48·65-s − 0.702·73-s + 0.433·85-s − 1.05·89-s + 1.82·97-s − 0.199·101-s − 0.574·109-s + 1.31·113-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: $\chi_{576} (1, \cdot )$
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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
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

−10.23383425889508895415100406682, −9.468761705895534248392912886381, −8.439450125780651634799093893382, −7.52766744709985624839545780760, −6.93944627141196287077129229697, −5.54216083419874931368270995905, −4.56897038264081606692544495668, −3.55808048734848453068190211023, −2.19360926670427462073525005533, 0, 2.19360926670427462073525005533, 3.55808048734848453068190211023, 4.56897038264081606692544495668, 5.54216083419874931368270995905, 6.93944627141196287077129229697, 7.52766744709985624839545780760, 8.439450125780651634799093893382, 9.468761705895534248392912886381, 10.23383425889508895415100406682

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