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Properties

Label 2-810-45.14-c2-0-21
Degree 22
Conductor 810810
Sign 0.522−0.852i0.522 - 0.852i
Analytic cond. 22.070922.0709
Root an. cond. 4.697964.69796
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.999 + 1.73i)4-s + (−4.89 − 0.997i)5-s + (−0.704 + 0.406i)7-s − 2.82·8-s + (−2.24 − 6.70i)10-s + (13.3 − 7.68i)11-s + (5.10 + 2.94i)13-s + (−0.996 − 0.575i)14-s + (−2.00 − 3.46i)16-s − 12.8·17-s + 1.24·19-s + (6.62 − 7.48i)20-s + (18.8 + 10.8i)22-s + (2.39 − 4.15i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.979 − 0.199i)5-s + (−0.100 + 0.0581i)7-s − 0.353·8-s + (−0.224 − 0.670i)10-s + (1.21 − 0.698i)11-s + (0.392 + 0.226i)13-s + (−0.0711 − 0.0410i)14-s + (−0.125 − 0.216i)16-s − 0.758·17-s + 0.0654·19-s + (0.331 − 0.374i)20-s + (0.855 + 0.494i)22-s + (0.104 − 0.180i)23-s + ⋯

Functional equation

Λ(s)=(810s/2 ΓC(s) L(s)=((0.522−0.852i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(810s/2 ΓC(s+1) L(s)=((0.522−0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 810810    =    2⋅34⋅52 \cdot 3^{4} \cdot 5
Sign: 0.522−0.852i0.522 - 0.852i
Analytic conductor: 22.070922.0709
Root analytic conductor: 4.697964.69796
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ810(539,⋅)\chi_{810} (539, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 810, ( :1), 0.522−0.852i)(2,\ 810,\ (\ :1),\ 0.522 - 0.852i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.8214629391.821462939
L(12)L(\frac12) ≈\approx 1.8214629391.821462939
L(2)L(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+(−0.707−1.22i)T 1 + (-0.707 - 1.22i)T
3 1 1
5 1+(4.89+0.997i)T 1 + (4.89 + 0.997i)T
good7 1+(0.704−0.406i)T+(24.5−42.4i)T2 1 + (0.704 - 0.406i)T + (24.5 - 42.4i)T^{2}
11 1+(−13.3+7.68i)T+(60.5−104.i)T2 1 + (-13.3 + 7.68i)T + (60.5 - 104. i)T^{2}
13 1+(−5.10−2.94i)T+(84.5+146.i)T2 1 + (-5.10 - 2.94i)T + (84.5 + 146. i)T^{2}
17 1+12.8T+289T2 1 + 12.8T + 289T^{2}
19 1−1.24T+361T2 1 - 1.24T + 361T^{2}
23 1+(−2.39+4.15i)T+(−264.5−458.i)T2 1 + (-2.39 + 4.15i)T + (-264.5 - 458. i)T^{2}
29 1+(−36.9+21.3i)T+(420.5−728.i)T2 1 + (-36.9 + 21.3i)T + (420.5 - 728. i)T^{2}
31 1+(2.10−3.64i)T+(−480.5−832.i)T2 1 + (2.10 - 3.64i)T + (-480.5 - 832. i)T^{2}
37 1−70.3iT−1.36e3T2 1 - 70.3iT - 1.36e3T^{2}
41 1+(6.09+3.52i)T+(840.5+1.45e3i)T2 1 + (6.09 + 3.52i)T + (840.5 + 1.45e3i)T^{2}
43 1+(−35.5+20.5i)T+(924.5−1.60e3i)T2 1 + (-35.5 + 20.5i)T + (924.5 - 1.60e3i)T^{2}
47 1+(−39.8−69.1i)T+(−1.10e3+1.91e3i)T2 1 + (-39.8 - 69.1i)T + (-1.10e3 + 1.91e3i)T^{2}
53 1−63.7T+2.80e3T2 1 - 63.7T + 2.80e3T^{2}
59 1+(−31.7−18.3i)T+(1.74e3+3.01e3i)T2 1 + (-31.7 - 18.3i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(−41.4−71.8i)T+(−1.86e3+3.22e3i)T2 1 + (-41.4 - 71.8i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(−77.0−44.5i)T+(2.24e3+3.88e3i)T2 1 + (-77.0 - 44.5i)T + (2.24e3 + 3.88e3i)T^{2}
71 1+69.6iT−5.04e3T2 1 + 69.6iT - 5.04e3T^{2}
73 1−89.6iT−5.32e3T2 1 - 89.6iT - 5.32e3T^{2}
79 1+(67.0+116.i)T+(−3.12e3+5.40e3i)T2 1 + (67.0 + 116. i)T + (-3.12e3 + 5.40e3i)T^{2}
83 1+(54.5+94.4i)T+(−3.44e3+5.96e3i)T2 1 + (54.5 + 94.4i)T + (-3.44e3 + 5.96e3i)T^{2}
89 1+137.iT−7.92e3T2 1 + 137. iT - 7.92e3T^{2}
97 1+(−78.4+45.3i)T+(4.70e3−8.14e3i)T2 1 + (-78.4 + 45.3i)T + (4.70e3 - 8.14e3i)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

−10.16288422018276912849846846285, −8.806163640978881807652917838130, −8.669184985757916299919219421863, −7.53438473828148498129463437883, −6.65766450585646688683355993329, −5.98209471227914911359373996108, −4.60993468227496691120267974576, −4.02335260674913528056436148356, −2.95164571100874786191989904165, −0.932099603229460782758466169489, 0.78493788701566893907115934662, 2.27139735458653082975685804872, 3.62847622748312572550362252916, 4.14936331237756148033578250081, 5.23097645726842980925037425562, 6.56665240917789391788096923166, 7.15045800524744369369433346501, 8.389071179396345288084563587347, 9.091510287179250376346346080968, 10.05936926156666934768444308888

Graph of the ZZ-function along the critical line