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Properties

Label 2-810-45.14-c2-0-8
Degree 22
Conductor 810810
Sign −0.930+0.366i-0.930 + 0.366i
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 + (3.31 + 3.74i)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 + (−9.79 + 1.99i)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 + (0.662 + 0.748i)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 + (−0.489 + 0.0997i)20-s + (−0.855 − 0.494i)22-s + (0.104 − 0.180i)23-s + ⋯

Functional equation

Λ(s)=(810s/2 ΓC(s) L(s)=((−0.930+0.366i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(810s/2 ΓC(s+1) L(s)=((−0.930+0.366i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 810810    =    2⋅34⋅52 \cdot 3^{4} \cdot 5
Sign: −0.930+0.366i-0.930 + 0.366i
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), −0.930+0.366i)(2,\ 810,\ (\ :1),\ -0.930 + 0.366i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.0470641321.047064132
L(12)L(\frac12) ≈\approx 1.0470641321.047064132
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+(−3.31−3.74i)T 1 + (-3.31 - 3.74i)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

−10.56278028648636027595409259857, −9.644719276300861632386803953870, −8.839374195868927835822493049328, −7.50608927319200846947326510141, −7.27560470720653643265265593444, −6.08073331269603452922736376154, −5.35825582067984182343999032342, −4.39623028454586611680212126169, −3.03548737831584495105297620840, −2.09138761141746423621700220638, 0.28349056199335134225424521840, 1.81074646093997913882376712388, 2.76688406007992074523015923868, 4.10804013910753723116328626251, 5.16157884552744510501618558804, 5.65342936061646373282582786533, 6.83965098024756968272827542006, 8.139633129792236423988145556417, 8.797536169127615290978025921770, 9.765495116943966444907509203409

Graph of the ZZ-function along the critical line