Properties

Label 2-648-9.4-c3-0-14
Degree $2$
Conductor $648$
Sign $-0.173 - 0.984i$
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  + (−7 + 12.1i)5-s + (12 + 20.7i)7-s + (14 + 24.2i)11-s + (37 − 64.0i)13-s + 82·17-s + 92·19-s + (−4 + 6.92i)23-s + (−35.5 − 61.4i)25-s + (69 + 119. i)29-s + (−40 + 69.2i)31-s − 336·35-s + 30·37-s + (−141 + 244. i)41-s + (−2 − 3.46i)43-s + (−120 − 207. i)47-s + ⋯
L(s)  = 1  + (−0.626 + 1.08i)5-s + (0.647 + 1.12i)7-s + (0.383 + 0.664i)11-s + (0.789 − 1.36i)13-s + 1.16·17-s + 1.11·19-s + (−0.0362 + 0.0628i)23-s + (−0.284 − 0.491i)25-s + (0.441 + 0.765i)29-s + (−0.231 + 0.401i)31-s − 1.62·35-s + 0.133·37-s + (−0.537 + 0.930i)41-s + (−0.00709 − 0.0122i)43-s + (−0.372 − 0.645i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.173 - 0.984i$
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: \((2,\ 648,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.074421594\)
\(L(\frac12)\) \(\approx\) \(2.074421594\)
\(L(\frac{5}{2})\) 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 \( 1 + (7 - 12.1i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-12 - 20.7i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-14 - 24.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-37 + 64.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 82T + 4.91e3T^{2} \)
19 \( 1 - 92T + 6.85e3T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-69 - 119. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (40 - 69.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 30T + 5.06e4T^{2} \)
41 \( 1 + (141 - 244. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (120 + 207. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 130T + 1.48e5T^{2} \)
59 \( 1 + (298 - 516. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-109 - 188. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-218 + 377. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 856T + 3.57e5T^{2} \)
73 \( 1 + 998T + 3.89e5T^{2} \)
79 \( 1 + (-16 - 27.7i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-754 - 1.30e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 246T + 7.04e5T^{2} \)
97 \( 1 + (433 + 749. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.48023480195193242931742421118, −9.625239550506241337803110864179, −8.468341391223881290872138516050, −7.80767048757835969635895123831, −6.96611293003743937221993075798, −5.81264931866387607073584964152, −5.05878181254525299583969326866, −3.51578422186503612533377845469, −2.86386113368538552078314028293, −1.32102376172182133613646497366, 0.71017897440555711550347109691, 1.45901289996908664180369986921, 3.55875316649750929410341632960, 4.24040063107997487079823841393, 5.14451637932726973327170862026, 6.34509132500831454100560780044, 7.49368367968520898836874632575, 8.126658211513499946455287313376, 8.961707295822698353477469615498, 9.819928736160054871634982316948

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