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Properties

Label 2-54-27.11-c2-0-4
Degree 22
Conductor 5454
Sign −0.0495+0.998i-0.0495 + 0.998i
Analytic cond. 1.471391.47139
Root an. cond. 1.213011.21301
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  + (0.909 − 1.08i)2-s + (−2.95 − 0.512i)3-s + (−0.347 − 1.96i)4-s + (2.71 − 7.45i)5-s + (−3.24 + 2.73i)6-s + (0.0787 − 0.446i)7-s + (−2.44 − 1.41i)8-s + (8.47 + 3.02i)9-s + (−5.60 − 9.71i)10-s + (4.82 + 13.2i)11-s + (0.0171 + 5.99i)12-s + (−9.80 + 8.22i)13-s + (−0.412 − 0.491i)14-s + (−11.8 + 20.6i)15-s + (−3.75 + 1.36i)16-s + (28.5 − 16.4i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (−0.985 − 0.170i)3-s + (−0.0868 − 0.492i)4-s + (0.542 − 1.49i)5-s + (−0.540 + 0.456i)6-s + (0.0112 − 0.0638i)7-s + (−0.306 − 0.176i)8-s + (0.941 + 0.336i)9-s + (−0.560 − 0.971i)10-s + (0.438 + 1.20i)11-s + (0.00143 + 0.499i)12-s + (−0.754 + 0.632i)13-s + (−0.0294 − 0.0351i)14-s + (−0.789 + 1.37i)15-s + (−0.234 + 0.0855i)16-s + (1.67 − 0.969i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((−0.0495+0.998i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0495 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((−0.0495+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0495 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: −0.0495+0.998i-0.0495 + 0.998i
Analytic conductor: 1.471391.47139
Root analytic conductor: 1.213011.21301
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ54(11,⋅)\chi_{54} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1), −0.0495+0.998i)(2,\ 54,\ (\ :1),\ -0.0495 + 0.998i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.801091−0.841840i0.801091 - 0.841840i
L(12)L(\frac12) ≈\approx 0.801091−0.841840i0.801091 - 0.841840i
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+(−0.909+1.08i)T 1 + (-0.909 + 1.08i)T
3 1+(2.95+0.512i)T 1 + (2.95 + 0.512i)T
good5 1+(−2.71+7.45i)T+(−19.1−16.0i)T2 1 + (-2.71 + 7.45i)T + (-19.1 - 16.0i)T^{2}
7 1+(−0.0787+0.446i)T+(−46.0−16.7i)T2 1 + (-0.0787 + 0.446i)T + (-46.0 - 16.7i)T^{2}
11 1+(−4.82−13.2i)T+(−92.6+77.7i)T2 1 + (-4.82 - 13.2i)T + (-92.6 + 77.7i)T^{2}
13 1+(9.80−8.22i)T+(29.3−166.i)T2 1 + (9.80 - 8.22i)T + (29.3 - 166. i)T^{2}
17 1+(−28.5+16.4i)T+(144.5−250.i)T2 1 + (-28.5 + 16.4i)T + (144.5 - 250. i)T^{2}
19 1+(−0.202+0.351i)T+(−180.5−312.i)T2 1 + (-0.202 + 0.351i)T + (-180.5 - 312. i)T^{2}
23 1+(−14.2+2.51i)T+(497.−180.i)T2 1 + (-14.2 + 2.51i)T + (497. - 180. i)T^{2}
29 1+(16.8−20.0i)T+(−146.−828.i)T2 1 + (16.8 - 20.0i)T + (-146. - 828. i)T^{2}
31 1+(−4.33−24.6i)T+(−903.+328.i)T2 1 + (-4.33 - 24.6i)T + (-903. + 328. i)T^{2}
37 1+(−3.84−6.65i)T+(−684.5+1.18e3i)T2 1 + (-3.84 - 6.65i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(15.9+18.9i)T+(−291.+1.65e3i)T2 1 + (15.9 + 18.9i)T + (-291. + 1.65e3i)T^{2}
43 1+(16.8−6.13i)T+(1.41e3−1.18e3i)T2 1 + (16.8 - 6.13i)T + (1.41e3 - 1.18e3i)T^{2}
47 1+(46.7+8.24i)T+(2.07e3+755.i)T2 1 + (46.7 + 8.24i)T + (2.07e3 + 755. i)T^{2}
53 1−0.261iT−2.80e3T2 1 - 0.261iT - 2.80e3T^{2}
59 1+(−18.8+51.7i)T+(−2.66e3−2.23e3i)T2 1 + (-18.8 + 51.7i)T + (-2.66e3 - 2.23e3i)T^{2}
61 1+(18.1−103.i)T+(−3.49e3−1.27e3i)T2 1 + (18.1 - 103. i)T + (-3.49e3 - 1.27e3i)T^{2}
67 1+(49.4−41.5i)T+(779.−4.42e3i)T2 1 + (49.4 - 41.5i)T + (779. - 4.42e3i)T^{2}
71 1+(94.6−54.6i)T+(2.52e3−4.36e3i)T2 1 + (94.6 - 54.6i)T + (2.52e3 - 4.36e3i)T^{2}
73 1+(−31.4+54.5i)T+(−2.66e3−4.61e3i)T2 1 + (-31.4 + 54.5i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(14.7+12.3i)T+(1.08e3+6.14e3i)T2 1 + (14.7 + 12.3i)T + (1.08e3 + 6.14e3i)T^{2}
83 1+(−36.7+43.7i)T+(−1.19e3−6.78e3i)T2 1 + (-36.7 + 43.7i)T + (-1.19e3 - 6.78e3i)T^{2}
89 1+(−89.7−51.8i)T+(3.96e3+6.85e3i)T2 1 + (-89.7 - 51.8i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(52.8−19.2i)T+(7.20e3−6.04e3i)T2 1 + (52.8 - 19.2i)T + (7.20e3 - 6.04e3i)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

−14.63659495007173232695284856805, −13.30975533988698384722493862788, −12.30444002368090790684641842240, −11.91472224811346262774028532794, −10.12242134535871983312392153510, −9.292237550333178996523448569262, −7.15083503525383830451260416508, −5.38485266564028263576716573466, −4.64240913726474890267412423156, −1.41090898549712650034754932274, 3.40575499655363090947144870374, 5.58580739548615630688803097536, 6.35309309687222240865104765000, 7.66851459705219236856100251635, 9.848784475471328167653082518452, 10.83710250122808227529506411666, 11.92125065442921931289912108490, 13.28492442479197956528118945505, 14.53930511992143102586237651661, 15.17380170452530639676301767123

Graph of the ZZ-function along the critical line