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Properties

Label 2-54-27.2-c2-0-5
Degree 22
Conductor 5454
Sign −0.974+0.224i-0.974 + 0.224i
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  + (−1.39 − 0.245i)2-s + (−2.10 + 2.13i)3-s + (1.87 + 0.684i)4-s + (−5.37 − 6.40i)5-s + (3.45 − 2.45i)6-s + (−8.61 + 3.13i)7-s + (−2.44 − 1.41i)8-s + (−0.136 − 8.99i)9-s + (5.91 + 10.2i)10-s + (−4.22 + 5.04i)11-s + (−5.41 + 2.57i)12-s + (1.04 + 5.91i)13-s + (12.7 − 2.25i)14-s + (25.0 + 1.99i)15-s + (3.06 + 2.57i)16-s + (−0.880 + 0.508i)17-s + ⋯
L(s)  = 1  + (−0.696 − 0.122i)2-s + (−0.701 + 0.712i)3-s + (0.469 + 0.171i)4-s + (−1.07 − 1.28i)5-s + (0.576 − 0.409i)6-s + (−1.23 + 0.448i)7-s + (−0.306 − 0.176i)8-s + (−0.0151 − 0.999i)9-s + (0.591 + 1.02i)10-s + (−0.384 + 0.458i)11-s + (−0.451 + 0.214i)12-s + (0.0802 + 0.455i)13-s + (0.912 − 0.160i)14-s + (1.66 + 0.133i)15-s + (0.191 + 0.160i)16-s + (−0.0517 + 0.0298i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((−0.974+0.224i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((−0.974+0.224i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: −0.974+0.224i-0.974 + 0.224i
Analytic conductor: 1.471391.47139
Root analytic conductor: 1.213011.21301
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ54(29,⋅)\chi_{54} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1), −0.974+0.224i)(2,\ 54,\ (\ :1),\ -0.974 + 0.224i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.00539746−0.0474864i0.00539746 - 0.0474864i
L(12)L(\frac12) ≈\approx 0.00539746−0.0474864i0.00539746 - 0.0474864i
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+(1.39+0.245i)T 1 + (1.39 + 0.245i)T
3 1+(2.10−2.13i)T 1 + (2.10 - 2.13i)T
good5 1+(5.37+6.40i)T+(−4.34+24.6i)T2 1 + (5.37 + 6.40i)T + (-4.34 + 24.6i)T^{2}
7 1+(8.61−3.13i)T+(37.5−31.4i)T2 1 + (8.61 - 3.13i)T + (37.5 - 31.4i)T^{2}
11 1+(4.22−5.04i)T+(−21.0−119.i)T2 1 + (4.22 - 5.04i)T + (-21.0 - 119. i)T^{2}
13 1+(−1.04−5.91i)T+(−158.+57.8i)T2 1 + (-1.04 - 5.91i)T + (-158. + 57.8i)T^{2}
17 1+(0.880−0.508i)T+(144.5−250.i)T2 1 + (0.880 - 0.508i)T + (144.5 - 250. i)T^{2}
19 1+(−10.5+18.2i)T+(−180.5−312.i)T2 1 + (-10.5 + 18.2i)T + (-180.5 - 312. i)T^{2}
23 1+(7.33−20.1i)T+(−405.−340.i)T2 1 + (7.33 - 20.1i)T + (-405. - 340. i)T^{2}
29 1+(47.7+8.41i)T+(790.+287.i)T2 1 + (47.7 + 8.41i)T + (790. + 287. i)T^{2}
31 1+(6.20+2.25i)T+(736.+617.i)T2 1 + (6.20 + 2.25i)T + (736. + 617. i)T^{2}
37 1+(33.2+57.6i)T+(−684.5+1.18e3i)T2 1 + (33.2 + 57.6i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(−52.2+9.20i)T+(1.57e3−574.i)T2 1 + (-52.2 + 9.20i)T + (1.57e3 - 574. i)T^{2}
43 1+(36.3+30.5i)T+(321.+1.82e3i)T2 1 + (36.3 + 30.5i)T + (321. + 1.82e3i)T^{2}
47 1+(2.23+6.13i)T+(−1.69e3+1.41e3i)T2 1 + (2.23 + 6.13i)T + (-1.69e3 + 1.41e3i)T^{2}
53 1−39.7iT−2.80e3T2 1 - 39.7iT - 2.80e3T^{2}
59 1+(18.0+21.4i)T+(−604.+3.42e3i)T2 1 + (18.0 + 21.4i)T + (-604. + 3.42e3i)T^{2}
61 1+(−38.6+14.0i)T+(2.85e3−2.39e3i)T2 1 + (-38.6 + 14.0i)T + (2.85e3 - 2.39e3i)T^{2}
67 1+(2.57+14.6i)T+(−4.21e3+1.53e3i)T2 1 + (2.57 + 14.6i)T + (-4.21e3 + 1.53e3i)T^{2}
71 1+(65.6−37.9i)T+(2.52e3−4.36e3i)T2 1 + (65.6 - 37.9i)T + (2.52e3 - 4.36e3i)T^{2}
73 1+(26.5−45.9i)T+(−2.66e3−4.61e3i)T2 1 + (26.5 - 45.9i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(14.2−80.6i)T+(−5.86e3−2.13e3i)T2 1 + (14.2 - 80.6i)T + (-5.86e3 - 2.13e3i)T^{2}
83 1+(−46.2−8.14i)T+(6.47e3+2.35e3i)T2 1 + (-46.2 - 8.14i)T + (6.47e3 + 2.35e3i)T^{2}
89 1+(−20.2−11.7i)T+(3.96e3+6.85e3i)T2 1 + (-20.2 - 11.7i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(−26.8−22.4i)T+(1.63e3+9.26e3i)T2 1 + (-26.8 - 22.4i)T + (1.63e3 + 9.26e3i)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

−15.39272962771982961042377731313, −12.93641441191020690626029479923, −12.12686911878517533131974200275, −11.17620411396599796302021988166, −9.579009143745517561189681981839, −8.991164600612885149918788491906, −7.28546357094058625613403430664, −5.48636438132931184867580122558, −3.85005782524118181190500196780, −0.06063835335364426742464483240, 3.24262616861434414148522131745, 6.12399087120482399586611543335, 7.11019968248089393826613717908, 7.999414546603424601404284804740, 10.13497339378946540719101114062, 10.92139577924473397736773857439, 11.97956887729132234336373694856, 13.22907657519646971922237598288, 14.70035817732095885545443332901, 16.00697145207135820479350126338

Graph of the ZZ-function along the critical line