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Properties

Label 2-648-9.7-c3-0-21
Degree $2$
Conductor $648$
Sign $0.766 + 0.642i$
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  + (−0.5 − 0.866i)5-s + (4.5 − 7.79i)7-s + (8.5 − 14.7i)11-s + (22 + 38.1i)13-s + 56·17-s − 94·19-s + (25 + 43.3i)23-s + (62 − 107. i)25-s + (15 − 25.9i)29-s + (69.5 + 120. i)31-s − 9·35-s − 174·37-s + (−159 − 275. i)41-s + (121 − 209. i)43-s + (315 − 545. i)47-s + ⋯
L(s)  = 1  + (−0.0447 − 0.0774i)5-s + (0.242 − 0.420i)7-s + (0.232 − 0.403i)11-s + (0.469 + 0.812i)13-s + 0.798·17-s − 1.13·19-s + (0.226 + 0.392i)23-s + (0.495 − 0.859i)25-s + (0.0960 − 0.166i)29-s + (0.402 + 0.697i)31-s − 0.0434·35-s − 0.773·37-s + (−0.605 − 1.04i)41-s + (0.429 − 0.743i)43-s + (0.977 − 1.69i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(38.2332\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :3/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.034972850\)
\(L(\frac12)\) \(\approx\) \(2.034972850\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-4.5 + 7.79i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-8.5 + 14.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-22 - 38.1i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 56T + 4.91e3T^{2} \)
19 \( 1 + 94T + 6.85e3T^{2} \)
23 \( 1 + (-25 - 43.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-15 + 25.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-69.5 - 120. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 174T + 5.06e4T^{2} \)
41 \( 1 + (159 + 275. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-121 + 209. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-315 + 545. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 547T + 1.48e5T^{2} \)
59 \( 1 + (-118 - 204. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (164 - 284. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (307 + 531. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 296T + 3.57e5T^{2} \)
73 \( 1 - 433T + 3.89e5T^{2} \)
79 \( 1 + (-28 + 48.4i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-612.5 + 1.06e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + (695.5 - 1.20e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.31270841299359271001445340525, −8.957468640339384850044338363425, −8.520732993696090944980361864126, −7.34092685399945128429836692587, −6.55250354443886904671462532600, −5.52541968764019026677602434661, −4.37861697391446849208278864444, −3.52496146207677907891953689133, −2.03277376476636519077455566515, −0.72070969346518296623381013508, 1.04325352669678325961741009889, 2.43632148183204619113348962650, 3.59567857369362876672730801214, 4.76811257889424113286989721148, 5.73293128425588619823500044338, 6.64525605510942042079646025120, 7.72608833448008049945241962559, 8.482057063421762127503116811308, 9.351457115277334640571542132617, 10.34047959431419332825595810828

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