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Properties

Label 2-847-77.16-c1-0-5
Degree 22
Conductor 847847
Sign −0.957+0.288i-0.957 + 0.288i
Analytic cond. 6.763326.76332
Root an. cond. 2.600642.60064
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  + (−1.49 − 0.318i)2-s + (−0.300 + 2.86i)3-s + (0.317 + 0.141i)4-s + (1.57 + 1.74i)5-s + (1.36 − 4.19i)6-s + (0.680 − 2.55i)7-s + (2.04 + 1.48i)8-s + (−5.17 − 1.10i)9-s + (−1.79 − 3.11i)10-s + (−0.500 + 0.866i)12-s + (0.0571 + 0.175i)13-s + (−1.83 + 3.61i)14-s + (−5.46 + 3.97i)15-s + (−3.06 − 3.39i)16-s + (−3.83 + 0.815i)17-s + (7.40 + 3.29i)18-s + ⋯
L(s)  = 1  + (−1.05 − 0.225i)2-s + (−0.173 + 1.65i)3-s + (0.158 + 0.0706i)4-s + (0.702 + 0.780i)5-s + (0.556 − 1.71i)6-s + (0.257 − 0.966i)7-s + (0.724 + 0.526i)8-s + (−1.72 − 0.366i)9-s + (−0.568 − 0.984i)10-s + (−0.144 + 0.249i)12-s + (0.0158 + 0.0487i)13-s + (−0.490 + 0.966i)14-s + (−1.41 + 1.02i)15-s + (−0.765 − 0.849i)16-s + (−0.930 + 0.197i)17-s + (1.74 + 0.777i)18-s + ⋯

Functional equation

Λ(s)=(847s/2 ΓC(s) L(s)=((−0.957+0.288i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(847s/2 ΓC(s+1/2) L(s)=((−0.957+0.288i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 847847    =    7⋅1127 \cdot 11^{2}
Sign: −0.957+0.288i-0.957 + 0.288i
Analytic conductor: 6.763326.76332
Root analytic conductor: 2.600642.60064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ847(632,⋅)\chi_{847} (632, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 847, ( :1/2), −0.957+0.288i)(2,\ 847,\ (\ :1/2),\ -0.957 + 0.288i)

Particular Values

L(1)L(1) ≈\approx 0.0604304−0.409982i0.0604304 - 0.409982i
L(12)L(\frac12) ≈\approx 0.0604304−0.409982i0.0604304 - 0.409982i
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)
bad7 1+(−0.680+2.55i)T 1 + (-0.680 + 2.55i)T
11 1 1
good2 1+(1.49+0.318i)T+(1.82+0.813i)T2 1 + (1.49 + 0.318i)T + (1.82 + 0.813i)T^{2}
3 1+(0.300−2.86i)T+(−2.93−0.623i)T2 1 + (0.300 - 2.86i)T + (-2.93 - 0.623i)T^{2}
5 1+(−1.57−1.74i)T+(−0.522+4.97i)T2 1 + (-1.57 - 1.74i)T + (-0.522 + 4.97i)T^{2}
13 1+(−0.0571−0.175i)T+(−10.5+7.64i)T2 1 + (-0.0571 - 0.175i)T + (-10.5 + 7.64i)T^{2}
17 1+(3.83−0.815i)T+(15.5−6.91i)T2 1 + (3.83 - 0.815i)T + (15.5 - 6.91i)T^{2}
19 1+(0.706−0.314i)T+(12.7−14.1i)T2 1 + (0.706 - 0.314i)T + (12.7 - 14.1i)T^{2}
23 1+(4.17−7.23i)T+(−11.5−19.9i)T2 1 + (4.17 - 7.23i)T + (-11.5 - 19.9i)T^{2}
29 1+(6.60−4.80i)T+(8.96−27.5i)T2 1 + (6.60 - 4.80i)T + (8.96 - 27.5i)T^{2}
31 1+(−1.77+1.97i)T+(−3.24−30.8i)T2 1 + (-1.77 + 1.97i)T + (-3.24 - 30.8i)T^{2}
37 1+(−0.713−6.78i)T+(−36.1+7.69i)T2 1 + (-0.713 - 6.78i)T + (-36.1 + 7.69i)T^{2}
41 1+(−0.344−0.250i)T+(12.6+38.9i)T2 1 + (-0.344 - 0.250i)T + (12.6 + 38.9i)T^{2}
43 1+1.18T+43T2 1 + 1.18T + 43T^{2}
47 1+(7.01−3.12i)T+(31.4−34.9i)T2 1 + (7.01 - 3.12i)T + (31.4 - 34.9i)T^{2}
53 1+(−4.38+4.87i)T+(−5.54−52.7i)T2 1 + (-4.38 + 4.87i)T + (-5.54 - 52.7i)T^{2}
59 1+(−0.186−0.0831i)T+(39.4+43.8i)T2 1 + (-0.186 - 0.0831i)T + (39.4 + 43.8i)T^{2}
61 1+(9.75+10.8i)T+(−6.37+60.6i)T2 1 + (9.75 + 10.8i)T + (-6.37 + 60.6i)T^{2}
67 1+(1.87+3.25i)T+(−33.5+58.0i)T2 1 + (1.87 + 3.25i)T + (-33.5 + 58.0i)T^{2}
71 1+(3.07−9.47i)T+(−57.4−41.7i)T2 1 + (3.07 - 9.47i)T + (-57.4 - 41.7i)T^{2}
73 1+(0.110+0.0490i)T+(48.8+54.2i)T2 1 + (0.110 + 0.0490i)T + (48.8 + 54.2i)T^{2}
79 1+(−0.320−0.0681i)T+(72.1+32.1i)T2 1 + (-0.320 - 0.0681i)T + (72.1 + 32.1i)T^{2}
83 1+(1.03−3.19i)T+(−67.1−48.7i)T2 1 + (1.03 - 3.19i)T + (-67.1 - 48.7i)T^{2}
89 1+(−2.32+4.02i)T+(−44.5−77.0i)T2 1 + (-2.32 + 4.02i)T + (-44.5 - 77.0i)T^{2}
97 1+(−4.05−12.4i)T+(−78.4+57.0i)T2 1 + (-4.05 - 12.4i)T + (-78.4 + 57.0i)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

−10.37052356231650429525268969653, −9.882966103030263597010682287989, −9.386857628654353229634427797922, −8.440241418466243637006847828841, −7.47067780230165797787912980551, −6.31551695619462075635085401583, −5.19243730187489844454062508307, −4.34794911463438874643957334815, −3.38143547481003515113879441075, −1.83308867339941717738977421069, 0.28951193368231981280083381486, 1.64119021970139209308348110458, 2.33619768482599554222863110998, 4.53341954234939125195878489427, 5.73203048234856648179718458796, 6.39294535186339766505502151983, 7.35766650823010949351015546032, 8.143138638006380178671449516744, 8.796005106302892121212897145466, 9.270592007230248979298608862805

Graph of the ZZ-function along the critical line