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Properties

Label 2-84-28.19-c3-0-16
Degree 22
Conductor 8484
Sign 0.801+0.597i0.801 + 0.597i
Analytic cond. 4.956164.95616
Root an. cond. 2.226242.22624
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0644 + 2.82i)2-s + (−1.5 − 2.59i)3-s + (−7.99 − 0.364i)4-s + (−4.32 − 2.49i)5-s + (7.44 − 4.07i)6-s + (15.5 − 10.1i)7-s + (1.54 − 22.5i)8-s + (−4.5 + 7.79i)9-s + (7.34 − 12.0i)10-s + (39.8 − 23.0i)11-s + (11.0 + 21.3i)12-s − 88.6i·13-s + (27.6 + 44.4i)14-s + 14.9i·15-s + (63.7 + 5.82i)16-s + (−86.9 + 50.2i)17-s + ⋯
L(s)  = 1  + (−0.0227 + 0.999i)2-s + (−0.288 − 0.499i)3-s + (−0.998 − 0.0455i)4-s + (−0.387 − 0.223i)5-s + (0.506 − 0.277i)6-s + (0.837 − 0.547i)7-s + (0.0682 − 0.997i)8-s + (−0.166 + 0.288i)9-s + (0.232 − 0.381i)10-s + (1.09 − 0.630i)11-s + (0.265 + 0.512i)12-s − 1.89i·13-s + (0.527 + 0.849i)14-s + 0.258i·15-s + (0.995 + 0.0909i)16-s + (−1.24 + 0.716i)17-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((0.801+0.597i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+3/2) L(s)=((0.801+0.597i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 0.801+0.597i0.801 + 0.597i
Analytic conductor: 4.956164.95616
Root analytic conductor: 2.226242.22624
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ84(19,⋅)\chi_{84} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :3/2), 0.801+0.597i)(2,\ 84,\ (\ :3/2),\ 0.801 + 0.597i)

Particular Values

L(2)L(2) ≈\approx 1.03830−0.344490i1.03830 - 0.344490i
L(12)L(\frac12) ≈\approx 1.03830−0.344490i1.03830 - 0.344490i
L(52)L(\frac{5}{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+(0.0644−2.82i)T 1 + (0.0644 - 2.82i)T
3 1+(1.5+2.59i)T 1 + (1.5 + 2.59i)T
7 1+(−15.5+10.1i)T 1 + (-15.5 + 10.1i)T
good5 1+(4.32+2.49i)T+(62.5+108.i)T2 1 + (4.32 + 2.49i)T + (62.5 + 108. i)T^{2}
11 1+(−39.8+23.0i)T+(665.5−1.15e3i)T2 1 + (-39.8 + 23.0i)T + (665.5 - 1.15e3i)T^{2}
13 1+88.6iT−2.19e3T2 1 + 88.6iT - 2.19e3T^{2}
17 1+(86.9−50.2i)T+(2.45e3−4.25e3i)T2 1 + (86.9 - 50.2i)T + (2.45e3 - 4.25e3i)T^{2}
19 1+(−4.47+7.75i)T+(−3.42e3−5.94e3i)T2 1 + (-4.47 + 7.75i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(−47.8−27.6i)T+(6.08e3+1.05e4i)T2 1 + (-47.8 - 27.6i)T + (6.08e3 + 1.05e4i)T^{2}
29 1−96.5T+2.43e4T2 1 - 96.5T + 2.43e4T^{2}
31 1+(125.+216.i)T+(−1.48e4+2.57e4i)T2 1 + (125. + 216. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(159.−276.i)T+(−2.53e4−4.38e4i)T2 1 + (159. - 276. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1−65.9iT−6.89e4T2 1 - 65.9iT - 6.89e4T^{2}
43 1+55.9iT−7.95e4T2 1 + 55.9iT - 7.95e4T^{2}
47 1+(−146.+253.i)T+(−5.19e4−8.99e4i)T2 1 + (-146. + 253. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−74.3−128.i)T+(−7.44e4+1.28e5i)T2 1 + (-74.3 - 128. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(−119.−206.i)T+(−1.02e5+1.77e5i)T2 1 + (-119. - 206. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−597.−345.i)T+(1.13e5+1.96e5i)T2 1 + (-597. - 345. i)T + (1.13e5 + 1.96e5i)T^{2}
67 1+(−575.+332.i)T+(1.50e5−2.60e5i)T2 1 + (-575. + 332. i)T + (1.50e5 - 2.60e5i)T^{2}
71 1−1.02e3iT−3.57e5T2 1 - 1.02e3iT - 3.57e5T^{2}
73 1+(−251.+145.i)T+(1.94e5−3.36e5i)T2 1 + (-251. + 145. i)T + (1.94e5 - 3.36e5i)T^{2}
79 1+(−173.−100.i)T+(2.46e5+4.26e5i)T2 1 + (-173. - 100. i)T + (2.46e5 + 4.26e5i)T^{2}
83 1+409.T+5.71e5T2 1 + 409.T + 5.71e5T^{2}
89 1+(−91.6−52.9i)T+(3.52e5+6.10e5i)T2 1 + (-91.6 - 52.9i)T + (3.52e5 + 6.10e5i)T^{2}
97 1−301.iT−9.12e5T2 1 - 301. iT - 9.12e5T^{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

−13.70893680388599281184943069742, −12.90496171802850656914060858423, −11.57495592660144492281948142302, −10.37630824860015266104019192760, −8.625080248123726576904053419786, −7.943611650104223358981928076660, −6.70547490595157215550773194466, −5.45647260129955161325193387711, −4.02690414081283509855569658911, −0.75840101505453992211111743762, 1.92220163136786090469645100770, 3.99122629532779565027976049628, 4.92718292136699267669818071871, 6.91421307914145690315073759486, 8.843963077009536935432684818266, 9.373688278973023697848210151558, 11.00659903403709215839227077481, 11.54973476362529443946993830548, 12.34252725443653622770873808165, 14.03618023729663770501319070448

Graph of the ZZ-function along the critical line