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Properties

Label 2-3e4-9.5-c8-0-28
Degree 22
Conductor 8181
Sign −0.342+0.939i-0.342 + 0.939i
Analytic cond. 32.997632.9976
Root an. cond. 5.744355.74435
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (25.4 − 14.6i)2-s + (304 − 526. i)4-s + (712. + 411. i)5-s + (−983.5 − 1.70e3i)7-s − 1.03e4i·8-s + 2.41e4·10-s + (−1.08e4 + 6.29e3i)11-s + (2.27e4 − 3.94e4i)13-s + (−5.00e4 − 2.89e4i)14-s + (−7.42e4 − 1.28e5i)16-s − 5.96e4i·17-s + 1.52e5·19-s + (4.33e5 − 2.50e5i)20-s + (−1.84e5 + 3.20e5i)22-s + (1.13e5 + 6.56e4i)23-s + ⋯
L(s)  = 1  + (1.59 − 0.918i)2-s + (1.18 − 2.05i)4-s + (1.14 + 0.658i)5-s + (−0.409 − 0.709i)7-s − 2.52i·8-s + 2.41·10-s + (−0.744 + 0.429i)11-s + (0.796 − 1.37i)13-s + (−1.30 − 0.752i)14-s + (−1.13 − 1.96i)16-s − 0.713i·17-s + 1.16·19-s + (2.70 − 1.56i)20-s + (−0.789 + 1.36i)22-s + (0.406 + 0.234i)23-s + ⋯

Functional equation

Λ(s)=(81s/2 ΓC(s) L(s)=((−0.342+0.939i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(81s/2 ΓC(s+4) L(s)=((−0.342+0.939i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8181    =    343^{4}
Sign: −0.342+0.939i-0.342 + 0.939i
Analytic conductor: 32.997632.9976
Root analytic conductor: 5.744355.74435
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ81(53,⋅)\chi_{81} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 81, ( :4), −0.342+0.939i)(2,\ 81,\ (\ :4),\ -0.342 + 0.939i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 3.40355−4.86078i3.40355 - 4.86078i
L(12)L(\frac12) ≈\approx 3.40355−4.86078i3.40355 - 4.86078i
L(5)L(5) 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)
bad3 1 1
good2 1+(−25.4+14.6i)T+(128−221.i)T2 1 + (-25.4 + 14.6i)T + (128 - 221. i)T^{2}
5 1+(−712.−411.i)T+(1.95e5+3.38e5i)T2 1 + (-712. - 411. i)T + (1.95e5 + 3.38e5i)T^{2}
7 1+(983.5+1.70e3i)T+(−2.88e6+4.99e6i)T2 1 + (983.5 + 1.70e3i)T + (-2.88e6 + 4.99e6i)T^{2}
11 1+(1.08e4−6.29e3i)T+(1.07e8−1.85e8i)T2 1 + (1.08e4 - 6.29e3i)T + (1.07e8 - 1.85e8i)T^{2}
13 1+(−2.27e4+3.94e4i)T+(−4.07e8−7.06e8i)T2 1 + (-2.27e4 + 3.94e4i)T + (-4.07e8 - 7.06e8i)T^{2}
17 1+5.96e4iT−6.97e9T2 1 + 5.96e4iT - 6.97e9T^{2}
19 1−1.52e5T+1.69e10T2 1 - 1.52e5T + 1.69e10T^{2}
23 1+(−1.13e5−6.56e4i)T+(3.91e10+6.78e10i)T2 1 + (-1.13e5 - 6.56e4i)T + (3.91e10 + 6.78e10i)T^{2}
29 1+(5.09e5−2.94e5i)T+(2.50e11−4.33e11i)T2 1 + (5.09e5 - 2.94e5i)T + (2.50e11 - 4.33e11i)T^{2}
31 1+(−8.21e4+1.42e5i)T+(−4.26e11−7.38e11i)T2 1 + (-8.21e4 + 1.42e5i)T + (-4.26e11 - 7.38e11i)T^{2}
37 1+6.63e5T+3.51e12T2 1 + 6.63e5T + 3.51e12T^{2}
41 1+(−8.12e5−4.69e5i)T+(3.99e12+6.91e12i)T2 1 + (-8.12e5 - 4.69e5i)T + (3.99e12 + 6.91e12i)T^{2}
43 1+(2.87e5+4.98e5i)T+(−5.84e12+1.01e13i)T2 1 + (2.87e5 + 4.98e5i)T + (-5.84e12 + 1.01e13i)T^{2}
47 1+(7.99e6−4.61e6i)T+(1.19e13−2.06e13i)T2 1 + (7.99e6 - 4.61e6i)T + (1.19e13 - 2.06e13i)T^{2}
53 1−1.03e7iT−6.22e13T2 1 - 1.03e7iT - 6.22e13T^{2}
59 1+(−4.36e6−2.51e6i)T+(7.34e13+1.27e14i)T2 1 + (-4.36e6 - 2.51e6i)T + (7.34e13 + 1.27e14i)T^{2}
61 1+(−9.60e6−1.66e7i)T+(−9.58e13+1.66e14i)T2 1 + (-9.60e6 - 1.66e7i)T + (-9.58e13 + 1.66e14i)T^{2}
67 1+(−2.99e5+5.17e5i)T+(−2.03e14−3.51e14i)T2 1 + (-2.99e5 + 5.17e5i)T + (-2.03e14 - 3.51e14i)T^{2}
71 1−2.92e7iT−6.45e14T2 1 - 2.92e7iT - 6.45e14T^{2}
73 1−1.28e7T+8.06e14T2 1 - 1.28e7T + 8.06e14T^{2}
79 1+(−1.17e7−2.04e7i)T+(−7.58e14+1.31e15i)T2 1 + (-1.17e7 - 2.04e7i)T + (-7.58e14 + 1.31e15i)T^{2}
83 1+(−2.89e7+1.67e7i)T+(1.12e15−1.95e15i)T2 1 + (-2.89e7 + 1.67e7i)T + (1.12e15 - 1.95e15i)T^{2}
89 1+2.82e7iT−3.93e15T2 1 + 2.82e7iT - 3.93e15T^{2}
97 1+(6.82e7+1.18e8i)T+(−3.91e15+6.78e15i)T2 1 + (6.82e7 + 1.18e8i)T + (-3.91e15 + 6.78e15i)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

−12.79050577280421776201627179869, −11.30431461026446242842591964433, −10.41571170682040141510629263326, −9.798896288008967586185600070921, −7.24386836750511443779637780404, −5.95775917723816383732082545175, −5.13146208090468048727092180639, −3.43524444481153067024501430182, −2.61336520083607999854862484340, −1.11637042076213741673761342588, 1.99399241660350366866713959910, 3.47718258096914453659246642738, 5.02203283637672483115106277881, 5.79737468239884109229724063311, 6.66325230025739260424999353779, 8.298272524065041022044410712408, 9.467241754996192991242606249924, 11.35316681000568789343291472602, 12.49481343785249503135280411659, 13.34450120501815180417643526288

Graph of the ZZ-function along the critical line