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Properties

Label 2-486-27.22-c1-0-2
Degree $2$
Conductor $486$
Sign $0.660 - 0.751i$
Analytic cond. $3.88072$
Root an. cond. $1.96995$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−2.97 + 1.08i)5-s + (0.640 + 3.62i)7-s + (−0.500 − 0.866i)8-s + (−1.58 + 2.74i)10-s + (2.14 + 0.781i)11-s + (2.37 + 1.99i)13-s + (2.82 + 2.36i)14-s + (−0.939 − 0.342i)16-s + (−0.862 + 1.49i)17-s + (1.69 + 2.93i)19-s + (0.550 + 3.11i)20-s + (2.14 − 0.781i)22-s + (−0.582 + 3.30i)23-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.0868 − 0.492i)4-s + (−1.33 + 0.484i)5-s + (0.241 + 1.37i)7-s + (−0.176 − 0.306i)8-s + (−0.500 + 0.867i)10-s + (0.647 + 0.235i)11-s + (0.658 + 0.552i)13-s + (0.754 + 0.633i)14-s + (−0.234 − 0.0855i)16-s + (−0.209 + 0.362i)17-s + (0.389 + 0.674i)19-s + (0.123 + 0.697i)20-s + (0.457 − 0.166i)22-s + (−0.121 + 0.688i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(486\)    =    \(2 \cdot 3^{5}\)
Sign: $0.660 - 0.751i$
Analytic conductor: \(3.88072\)
Root analytic conductor: \(1.96995\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{486} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 486,\ (\ :1/2),\ 0.660 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32462 + 0.599235i\)
\(L(\frac12)\) \(\approx\) \(1.32462 + 0.599235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 \)
good5 \( 1 + (2.97 - 1.08i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.640 - 3.62i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.14 - 0.781i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-2.37 - 1.99i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.862 - 1.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.69 - 2.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.582 - 3.30i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (0.448 - 0.376i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.805 - 4.56i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-3.65 + 6.32i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.42 + 4.55i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.55 + 0.567i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.668 + 3.79i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 2.58T + 53T^{2} \)
59 \( 1 + (-9.08 + 3.30i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.27 - 12.8i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (6.59 + 5.53i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.993 - 1.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.32 + 9.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.7 + 9.05i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (2.37 - 1.99i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-8.67 - 15.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.63 + 3.14i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.44944943484427709641563202784, −10.50355353152457175032667780496, −9.230071888598257404886858974621, −8.504815087349088431851338236389, −7.41375410141246244265582696323, −6.35830985035246254267110320025, −5.34973697778308917276230467192, −4.07771006931638911594929911236, −3.33919118712362210474060812204, −1.87265417972722959685767657143, 0.791841045048874283143138436927, 3.31785788122000719532085920832, 4.16291068253147656393727524307, 4.83442915973555680437701271915, 6.33706491222367243947835996051, 7.26363476554919407065428094576, 7.974907996648087821962344928217, 8.706032420460223997313125751778, 10.04471338621537046395935485812, 11.32697945364584684666313171656

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