Properties

Label 2-648-9.4-c3-0-9
Degree $2$
Conductor $648$
Sign $-0.939 - 0.342i$
Analytic cond. $38.2332$
Root an. cond. $6.18330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.70 + 9.88i)5-s + (14.9 + 25.8i)7-s + (33.1 + 57.3i)11-s + (−19.9 + 34.4i)13-s − 107.·17-s + 70.3·19-s + (3.45 − 5.99i)23-s + (−2.66 − 4.61i)25-s + (18.3 + 31.7i)29-s + (115. − 200. i)31-s − 340.·35-s + 36.8·37-s + (214. − 371. i)41-s + (37.1 + 64.3i)43-s + (26.2 + 45.5i)47-s + ⋯
L(s)  = 1  + (−0.510 + 0.884i)5-s + (0.805 + 1.39i)7-s + (0.907 + 1.57i)11-s + (−0.424 + 0.735i)13-s − 1.53·17-s + 0.849·19-s + (0.0313 − 0.0543i)23-s + (−0.0213 − 0.0369i)25-s + (0.117 + 0.203i)29-s + (0.670 − 1.16i)31-s − 1.64·35-s + 0.163·37-s + (0.817 − 1.41i)41-s + (0.131 + 0.228i)43-s + (0.0815 + 0.141i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(38.2332\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :3/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.653811506\)
\(L(\frac12)\) \(\approx\) \(1.653811506\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.70 - 9.88i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-14.9 - 25.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-33.1 - 57.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (19.9 - 34.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 107.T + 4.91e3T^{2} \)
19 \( 1 - 70.3T + 6.85e3T^{2} \)
23 \( 1 + (-3.45 + 5.99i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-18.3 - 31.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-115. + 200. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 36.8T + 5.06e4T^{2} \)
41 \( 1 + (-214. + 371. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-37.1 - 64.3i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-26.2 - 45.5i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 288.T + 1.48e5T^{2} \)
59 \( 1 + (391. - 678. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (219. + 380. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-109. + 189. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 790.T + 3.57e5T^{2} \)
73 \( 1 - 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + (219. + 380. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (25.4 + 44.0i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 719.T + 7.04e5T^{2} \)
97 \( 1 + (-599. - 1.03e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.69227319323717320094261822762, −9.395376552876622342338188142508, −9.058631924485708012428810828680, −7.77049450098842219308546252355, −7.04785972559845358109066781381, −6.21079044345563968752056599504, −4.88933674352806933577183375269, −4.15942588834132507394185912703, −2.59969729938978975763646384272, −1.84436103792039187991086322047, 0.51913460457170857065425030902, 1.23440485963593287961864539033, 3.19332706348158947710234427559, 4.27886286454434475181516457246, 4.88734942728512242051211416755, 6.19979711591800297982765225212, 7.23695718845750045039481295614, 8.173589186441300541863661449468, 8.657353998614367729531634211184, 9.744642583339052063364532390644

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