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Properties

Label 2-486-27.22-c1-0-2
Degree 22
Conductor 486486
Sign 0.660−0.751i0.660 - 0.751i
Analytic cond. 3.880723.88072
Root an. cond. 1.969951.96995
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(486s/2 ΓC(s) L(s)=((0.660−0.751i) Λ‾(2−s)\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}
Λ(s)=(486s/2 ΓC(s+1/2) L(s)=((0.660−0.751i) Λ‾(1−s)\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: 22
Conductor: 486486    =    2⋅352 \cdot 3^{5}
Sign: 0.660−0.751i0.660 - 0.751i
Analytic conductor: 3.880723.88072
Root analytic conductor: 1.969951.96995
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ486(109,⋅)\chi_{486} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 486, ( :1/2), 0.660−0.751i)(2,\ 486,\ (\ :1/2),\ 0.660 - 0.751i)

Particular Values

L(1)L(1) ≈\approx 1.32462+0.599235i1.32462 + 0.599235i
L(12)L(\frac12) ≈\approx 1.32462+0.599235i1.32462 + 0.599235i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
3 1 1
good5 1+(2.97−1.08i)T+(3.83−3.21i)T2 1 + (2.97 - 1.08i)T + (3.83 - 3.21i)T^{2}
7 1+(−0.640−3.62i)T+(−6.57+2.39i)T2 1 + (-0.640 - 3.62i)T + (-6.57 + 2.39i)T^{2}
11 1+(−2.14−0.781i)T+(8.42+7.07i)T2 1 + (-2.14 - 0.781i)T + (8.42 + 7.07i)T^{2}
13 1+(−2.37−1.99i)T+(2.25+12.8i)T2 1 + (-2.37 - 1.99i)T + (2.25 + 12.8i)T^{2}
17 1+(0.862−1.49i)T+(−8.5−14.7i)T2 1 + (0.862 - 1.49i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.69−2.93i)T+(−9.5+16.4i)T2 1 + (-1.69 - 2.93i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.582−3.30i)T+(−21.6−7.86i)T2 1 + (0.582 - 3.30i)T + (-21.6 - 7.86i)T^{2}
29 1+(0.448−0.376i)T+(5.03−28.5i)T2 1 + (0.448 - 0.376i)T + (5.03 - 28.5i)T^{2}
31 1+(0.805−4.56i)T+(−29.1−10.6i)T2 1 + (0.805 - 4.56i)T + (-29.1 - 10.6i)T^{2}
37 1+(−3.65+6.32i)T+(−18.5−32.0i)T2 1 + (-3.65 + 6.32i)T + (-18.5 - 32.0i)T^{2}
41 1+(5.42+4.55i)T+(7.11+40.3i)T2 1 + (5.42 + 4.55i)T + (7.11 + 40.3i)T^{2}
43 1+(1.55+0.567i)T+(32.9+27.6i)T2 1 + (1.55 + 0.567i)T + (32.9 + 27.6i)T^{2}
47 1+(0.668+3.79i)T+(−44.1+16.0i)T2 1 + (0.668 + 3.79i)T + (-44.1 + 16.0i)T^{2}
53 1+2.58T+53T2 1 + 2.58T + 53T^{2}
59 1+(−9.08+3.30i)T+(45.1−37.9i)T2 1 + (-9.08 + 3.30i)T + (45.1 - 37.9i)T^{2}
61 1+(−2.27−12.8i)T+(−57.3+20.8i)T2 1 + (-2.27 - 12.8i)T + (-57.3 + 20.8i)T^{2}
67 1+(6.59+5.53i)T+(11.6+65.9i)T2 1 + (6.59 + 5.53i)T + (11.6 + 65.9i)T^{2}
71 1+(0.993−1.72i)T+(−35.5−61.4i)T2 1 + (0.993 - 1.72i)T + (-35.5 - 61.4i)T^{2}
73 1+(5.32+9.22i)T+(−36.5+63.2i)T2 1 + (5.32 + 9.22i)T + (-36.5 + 63.2i)T^{2}
79 1+(−10.7+9.05i)T+(13.7−77.7i)T2 1 + (-10.7 + 9.05i)T + (13.7 - 77.7i)T^{2}
83 1+(2.37−1.99i)T+(14.4−81.7i)T2 1 + (2.37 - 1.99i)T + (14.4 - 81.7i)T^{2}
89 1+(−8.67−15.0i)T+(−44.5+77.0i)T2 1 + (-8.67 - 15.0i)T + (-44.5 + 77.0i)T^{2}
97 1+(8.63+3.14i)T+(74.3+62.3i)T2 1 + (8.63 + 3.14i)T + (74.3 + 62.3i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line