Properties

Label 2-24e2-4.3-c0-0-0
Degree $2$
Conductor $576$
Sign $1$
Analytic cond. $0.287461$
Root an. cond. $0.536154$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·13-s − 25-s − 2·37-s + 49-s − 2·61-s − 2·73-s − 2·97-s + 2·109-s + ⋯
L(s)  = 1  + 2·13-s − 25-s − 2·37-s + 49-s − 2·61-s − 2·73-s − 2·97-s + 2·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(0.287461\)
Root analytic conductor: \(0.536154\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{576} (127, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9485701337\)
\(L(\frac12)\) \(\approx\) \(0.9485701337\)
\(L(1)\) 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 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 + T )^{2} \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 + T )^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
97 \( ( 1 + T )^{2} \)
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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

−10.90771780233178235159130714392, −10.19366891545730683743532188681, −9.024509684863061469596680594047, −8.437926701374658738630645762120, −7.38608666175590124683562879224, −6.31086762823771710049583842357, −5.56406142798722163392950984421, −4.19712792058583127732173175389, −3.27725001056031716507892040372, −1.61093910315831831020049975359, 1.61093910315831831020049975359, 3.27725001056031716507892040372, 4.19712792058583127732173175389, 5.56406142798722163392950984421, 6.31086762823771710049583842357, 7.38608666175590124683562879224, 8.437926701374658738630645762120, 9.024509684863061469596680594047, 10.19366891545730683743532188681, 10.90771780233178235159130714392

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