Properties

Label 2-648-1.1-c3-0-15
Degree $2$
Conductor $648$
Sign $1$
Analytic cond. $38.2332$
Root an. cond. $6.18330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.33·5-s + 19.1·7-s + 41.2·11-s + 39.2·13-s + 59.9·17-s − 41.2·19-s − 53.8·23-s − 84.8·25-s − 65.2·29-s + 64.7·31-s + 121.·35-s + 293.·37-s + 207.·41-s − 377.·43-s + 323.·47-s + 25.4·49-s − 340.·53-s + 261.·55-s + 406.·59-s − 757.·61-s + 248.·65-s + 844.·67-s + 859.·71-s − 789.·73-s + 791.·77-s − 217.·79-s + 101.·83-s + ⋯
L(s)  = 1  + 0.567·5-s + 1.03·7-s + 1.13·11-s + 0.837·13-s + 0.855·17-s − 0.498·19-s − 0.487·23-s − 0.678·25-s − 0.417·29-s + 0.375·31-s + 0.587·35-s + 1.30·37-s + 0.789·41-s − 1.34·43-s + 1.00·47-s + 0.0741·49-s − 0.882·53-s + 0.641·55-s + 0.897·59-s − 1.58·61-s + 0.474·65-s + 1.53·67-s + 1.43·71-s − 1.26·73-s + 1.17·77-s − 0.310·79-s + 0.133·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(38.2332\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.916627198\)
\(L(\frac12)\) \(\approx\) \(2.916627198\)
\(L(\frac{5}{2})\) 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 \( 1 - 6.33T + 125T^{2} \)
7 \( 1 - 19.1T + 343T^{2} \)
11 \( 1 - 41.2T + 1.33e3T^{2} \)
13 \( 1 - 39.2T + 2.19e3T^{2} \)
17 \( 1 - 59.9T + 4.91e3T^{2} \)
19 \( 1 + 41.2T + 6.85e3T^{2} \)
23 \( 1 + 53.8T + 1.21e4T^{2} \)
29 \( 1 + 65.2T + 2.43e4T^{2} \)
31 \( 1 - 64.7T + 2.97e4T^{2} \)
37 \( 1 - 293.T + 5.06e4T^{2} \)
41 \( 1 - 207.T + 6.89e4T^{2} \)
43 \( 1 + 377.T + 7.95e4T^{2} \)
47 \( 1 - 323.T + 1.03e5T^{2} \)
53 \( 1 + 340.T + 1.48e5T^{2} \)
59 \( 1 - 406.T + 2.05e5T^{2} \)
61 \( 1 + 757.T + 2.26e5T^{2} \)
67 \( 1 - 844.T + 3.00e5T^{2} \)
71 \( 1 - 859.T + 3.57e5T^{2} \)
73 \( 1 + 789.T + 3.89e5T^{2} \)
79 \( 1 + 217.T + 4.93e5T^{2} \)
83 \( 1 - 101.T + 5.71e5T^{2} \)
89 \( 1 - 1.24e3T + 7.04e5T^{2} \)
97 \( 1 + 1.74e3T + 9.12e5T^{2} \)
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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

−10.09495890021566546209788882797, −9.314702381683978280278111849802, −8.400881553069678388059525695237, −7.65757063528332426388583359833, −6.39917802288384660339302714165, −5.75026719653185088725618698427, −4.56805206017389704820339630267, −3.61738216645024332781139291175, −2.03615425234351382178290067589, −1.11259601734816875843562038013, 1.11259601734816875843562038013, 2.03615425234351382178290067589, 3.61738216645024332781139291175, 4.56805206017389704820339630267, 5.75026719653185088725618698427, 6.39917802288384660339302714165, 7.65757063528332426388583359833, 8.400881553069678388059525695237, 9.314702381683978280278111849802, 10.09495890021566546209788882797

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