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Properties

Label 2-54-27.20-c2-0-3
Degree 22
Conductor 5454
Sign 0.989−0.143i0.989 - 0.143i
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  + (0.483 + 1.32i)2-s + (0.320 − 2.98i)3-s + (−1.53 + 1.28i)4-s + (5.90 − 1.04i)5-s + (4.11 − 1.01i)6-s + (5.59 + 4.69i)7-s + (−2.44 − 1.41i)8-s + (−8.79 − 1.91i)9-s + (4.23 + 7.34i)10-s + (−20.3 − 3.59i)11-s + (3.34 + 4.98i)12-s + (6.40 + 2.33i)13-s + (−3.53 + 9.71i)14-s + (−1.20 − 17.9i)15-s + (0.694 − 3.93i)16-s + (1.96 − 1.13i)17-s + ⋯
L(s)  = 1  + (0.241 + 0.664i)2-s + (0.106 − 0.994i)3-s + (−0.383 + 0.321i)4-s + (1.18 − 0.208i)5-s + (0.686 − 0.169i)6-s + (0.799 + 0.671i)7-s + (−0.306 − 0.176i)8-s + (−0.977 − 0.212i)9-s + (0.423 + 0.734i)10-s + (−1.85 − 0.326i)11-s + (0.278 + 0.415i)12-s + (0.492 + 0.179i)13-s + (−0.252 + 0.693i)14-s + (−0.0806 − 1.19i)15-s + (0.0434 − 0.246i)16-s + (0.115 − 0.0667i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.989−0.143i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((0.989−0.143i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.989−0.143i0.989 - 0.143i
Analytic conductor: 1.471391.47139
Root analytic conductor: 1.213011.21301
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ54(47,⋅)\chi_{54} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1), 0.989−0.143i)(2,\ 54,\ (\ :1),\ 0.989 - 0.143i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.42603+0.102725i1.42603 + 0.102725i
L(12)L(\frac12) ≈\approx 1.42603+0.102725i1.42603 + 0.102725i
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+(−0.483−1.32i)T 1 + (-0.483 - 1.32i)T
3 1+(−0.320+2.98i)T 1 + (-0.320 + 2.98i)T
good5 1+(−5.90+1.04i)T+(23.4−8.55i)T2 1 + (-5.90 + 1.04i)T + (23.4 - 8.55i)T^{2}
7 1+(−5.59−4.69i)T+(8.50+48.2i)T2 1 + (-5.59 - 4.69i)T + (8.50 + 48.2i)T^{2}
11 1+(20.3+3.59i)T+(113.+41.3i)T2 1 + (20.3 + 3.59i)T + (113. + 41.3i)T^{2}
13 1+(−6.40−2.33i)T+(129.+108.i)T2 1 + (-6.40 - 2.33i)T + (129. + 108. i)T^{2}
17 1+(−1.96+1.13i)T+(144.5−250.i)T2 1 + (-1.96 + 1.13i)T + (144.5 - 250. i)T^{2}
19 1+(12.1−21.0i)T+(−180.5−312.i)T2 1 + (12.1 - 21.0i)T + (-180.5 - 312. i)T^{2}
23 1+(15.5+18.5i)T+(−91.8+520.i)T2 1 + (15.5 + 18.5i)T + (-91.8 + 520. i)T^{2}
29 1+(−6.32−17.3i)T+(−644.+540.i)T2 1 + (-6.32 - 17.3i)T + (-644. + 540. i)T^{2}
31 1+(−16.6+13.9i)T+(166.−946.i)T2 1 + (-16.6 + 13.9i)T + (166. - 946. i)T^{2}
37 1+(−22.3−38.7i)T+(−684.5+1.18e3i)T2 1 + (-22.3 - 38.7i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(−17.5+48.1i)T+(−1.28e3−1.08e3i)T2 1 + (-17.5 + 48.1i)T + (-1.28e3 - 1.08e3i)T^{2}
43 1+(−6.41+36.3i)T+(−1.73e3−632.i)T2 1 + (-6.41 + 36.3i)T + (-1.73e3 - 632. i)T^{2}
47 1+(−8.15+9.71i)T+(−383.−2.17e3i)T2 1 + (-8.15 + 9.71i)T + (-383. - 2.17e3i)T^{2}
53 1−17.7iT−2.80e3T2 1 - 17.7iT - 2.80e3T^{2}
59 1+(−64.9+11.4i)T+(3.27e3−1.19e3i)T2 1 + (-64.9 + 11.4i)T + (3.27e3 - 1.19e3i)T^{2}
61 1+(−32.3−27.1i)T+(646.+3.66e3i)T2 1 + (-32.3 - 27.1i)T + (646. + 3.66e3i)T^{2}
67 1+(111.+40.4i)T+(3.43e3+2.88e3i)T2 1 + (111. + 40.4i)T + (3.43e3 + 2.88e3i)T^{2}
71 1+(46.1−26.6i)T+(2.52e3−4.36e3i)T2 1 + (46.1 - 26.6i)T + (2.52e3 - 4.36e3i)T^{2}
73 1+(−33.8+58.6i)T+(−2.66e3−4.61e3i)T2 1 + (-33.8 + 58.6i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(−104.+38.0i)T+(4.78e3−4.01e3i)T2 1 + (-104. + 38.0i)T + (4.78e3 - 4.01e3i)T^{2}
83 1+(−8.63−23.7i)T+(−5.27e3+4.42e3i)T2 1 + (-8.63 - 23.7i)T + (-5.27e3 + 4.42e3i)T^{2}
89 1+(35.4+20.4i)T+(3.96e3+6.85e3i)T2 1 + (35.4 + 20.4i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(−7.02+39.8i)T+(−8.84e3−3.21e3i)T2 1 + (-7.02 + 39.8i)T + (-8.84e3 - 3.21e3i)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

−14.91902165795005601438885060251, −13.87928894274968829674755140342, −13.17787487762470231260021246811, −12.13204730585545345598417985861, −10.44761319654681122853666920431, −8.673656772823837132699038281628, −7.895854629753285982683969700626, −6.14005312120654799924888898173, −5.34822171942748539850011891057, −2.26485979238134437549567390238, 2.54443760354670577729033765190, 4.55675581519798417268653461907, 5.70537321916241728598723569154, 8.076408782720419779323826776730, 9.646506546170726374302772556045, 10.46997567847221656283300427835, 11.18137352670377737396444068776, 13.14004942153981720913948077904, 13.82973466868233419891362729756, 14.92817190244522674129892465077

Graph of the ZZ-function along the critical line