Properties

Label 2-648-8.3-c0-0-0
Degree $2$
Conductor $648$
Sign $1$
Analytic cond. $0.323394$
Root an. cond. $0.568677$
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-s + 4-s − 8-s + 11-s + 16-s + 17-s − 19-s − 22-s + 25-s − 32-s − 34-s + 38-s + 41-s − 43-s + 44-s + 49-s − 50-s + 59-s + 64-s − 67-s + 68-s − 73-s − 76-s − 82-s − 2·83-s + 86-s − 88-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + 11-s + 16-s + 17-s − 19-s − 22-s + 25-s − 32-s − 34-s + 38-s + 41-s − 43-s + 44-s + 49-s − 50-s + 59-s + 64-s − 67-s + 68-s − 73-s − 76-s − 82-s − 2·83-s + 86-s − 88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(0.323394\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{648} (163, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6532441654\)
\(L(\frac12)\) \(\approx\) \(0.6532441654\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
good5 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 - T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 - T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 - T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 + T )^{2} \)
89 \( ( 1 + T )^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.60341475152256943221806619334, −9.874347180030868415081602803923, −8.982486746031989707936807894926, −8.351644822249025280774420685553, −7.29671137085968494935281080661, −6.54439724480456005171908411935, −5.59468296879334546829024752782, −4.07812634788854752097901427150, −2.81441074757590216582168505230, −1.36724421983330013271266289894, 1.36724421983330013271266289894, 2.81441074757590216582168505230, 4.07812634788854752097901427150, 5.59468296879334546829024752782, 6.54439724480456005171908411935, 7.29671137085968494935281080661, 8.351644822249025280774420685553, 8.982486746031989707936807894926, 9.874347180030868415081602803923, 10.60341475152256943221806619334

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