Properties

Label 2-648-1.1-c3-0-0
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  − 20.2·5-s − 24.7·7-s − 18.8·11-s − 48.4·13-s − 40.4·17-s + 7.82·19-s − 157.·23-s + 283.·25-s − 219.·29-s − 139.·31-s + 500.·35-s + 270.·37-s − 30.7·41-s + 57.2·43-s + 143.·47-s + 269.·49-s − 180.·53-s + 380.·55-s + 317.·59-s + 759.·61-s + 979.·65-s + 428.·67-s − 29.0·71-s − 327.·73-s + 465.·77-s − 1.02e3·79-s − 454.·83-s + ⋯
L(s)  = 1  − 1.80·5-s − 1.33·7-s − 0.515·11-s − 1.03·13-s − 0.577·17-s + 0.0945·19-s − 1.43·23-s + 2.27·25-s − 1.40·29-s − 0.807·31-s + 2.41·35-s + 1.20·37-s − 0.117·41-s + 0.203·43-s + 0.444·47-s + 0.784·49-s − 0.466·53-s + 0.932·55-s + 0.699·59-s + 1.59·61-s + 1.86·65-s + 0.780·67-s − 0.0486·71-s − 0.525·73-s + 0.688·77-s − 1.45·79-s − 0.601·83-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

\(L(2)\) \(\approx\) \(0.2104479364\)
\(L(\frac12)\) \(\approx\) \(0.2104479364\)
\(L(\frac{5}{2})\) 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 \( 1 + 20.2T + 125T^{2} \)
7 \( 1 + 24.7T + 343T^{2} \)
11 \( 1 + 18.8T + 1.33e3T^{2} \)
13 \( 1 + 48.4T + 2.19e3T^{2} \)
17 \( 1 + 40.4T + 4.91e3T^{2} \)
19 \( 1 - 7.82T + 6.85e3T^{2} \)
23 \( 1 + 157.T + 1.21e4T^{2} \)
29 \( 1 + 219.T + 2.43e4T^{2} \)
31 \( 1 + 139.T + 2.97e4T^{2} \)
37 \( 1 - 270.T + 5.06e4T^{2} \)
41 \( 1 + 30.7T + 6.89e4T^{2} \)
43 \( 1 - 57.2T + 7.95e4T^{2} \)
47 \( 1 - 143.T + 1.03e5T^{2} \)
53 \( 1 + 180.T + 1.48e5T^{2} \)
59 \( 1 - 317.T + 2.05e5T^{2} \)
61 \( 1 - 759.T + 2.26e5T^{2} \)
67 \( 1 - 428.T + 3.00e5T^{2} \)
71 \( 1 + 29.0T + 3.57e5T^{2} \)
73 \( 1 + 327.T + 3.89e5T^{2} \)
79 \( 1 + 1.02e3T + 4.93e5T^{2} \)
83 \( 1 + 454.T + 5.71e5T^{2} \)
89 \( 1 + 677.T + 7.04e5T^{2} \)
97 \( 1 + 397.T + 9.12e5T^{2} \)
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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

−10.11209974284139222031967402654, −9.340593131699343861399988939298, −8.254595207795435690412684933386, −7.49060728980552985655858032238, −6.85905380869545283292073324705, −5.61479140298633394280257191154, −4.31305110224365006134219748640, −3.62674809253531812832437024748, −2.54372213775187925322037033747, −0.24923286569574271303155644015, 0.24923286569574271303155644015, 2.54372213775187925322037033747, 3.62674809253531812832437024748, 4.31305110224365006134219748640, 5.61479140298633394280257191154, 6.85905380869545283292073324705, 7.49060728980552985655858032238, 8.254595207795435690412684933386, 9.340593131699343861399988939298, 10.11209974284139222031967402654

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