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Properties

Label 2-54-27.11-c2-0-1
Degree 22
Conductor 5454
Sign 0.999+0.0275i0.999 + 0.0275i
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 + (2.80 − 1.05i)3-s + (−0.347 − 1.96i)4-s + (2.86 − 7.86i)5-s + (−1.41 + 4.00i)6-s + (−1.95 + 11.0i)7-s + (2.44 + 1.41i)8-s + (6.77 − 5.92i)9-s + (5.92 + 10.2i)10-s + (0.538 + 1.47i)11-s + (−3.05 − 5.16i)12-s + (−6.82 + 5.72i)13-s + (−10.2 − 12.1i)14-s + (−0.256 − 25.1i)15-s + (−3.75 + 1.36i)16-s + (−16.9 + 9.80i)17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (0.936 − 0.351i)3-s + (−0.0868 − 0.492i)4-s + (0.572 − 1.57i)5-s + (−0.235 + 0.666i)6-s + (−0.278 + 1.58i)7-s + (0.306 + 0.176i)8-s + (0.752 − 0.658i)9-s + (0.592 + 1.02i)10-s + (0.0489 + 0.134i)11-s + (−0.254 − 0.430i)12-s + (−0.524 + 0.440i)13-s + (−0.729 − 0.869i)14-s + (−0.0170 − 1.67i)15-s + (−0.234 + 0.0855i)16-s + (−0.999 + 0.576i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.999+0.0275i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0275i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((0.999+0.0275i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0275i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.999+0.0275i0.999 + 0.0275i
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), 0.999+0.0275i)(2,\ 54,\ (\ :1),\ 0.999 + 0.0275i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.23893−0.0170708i1.23893 - 0.0170708i
L(12)L(\frac12) ≈\approx 1.23893−0.0170708i1.23893 - 0.0170708i
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+(−2.80+1.05i)T 1 + (-2.80 + 1.05i)T
good5 1+(−2.86+7.86i)T+(−19.1−16.0i)T2 1 + (-2.86 + 7.86i)T + (-19.1 - 16.0i)T^{2}
7 1+(1.95−11.0i)T+(−46.0−16.7i)T2 1 + (1.95 - 11.0i)T + (-46.0 - 16.7i)T^{2}
11 1+(−0.538−1.47i)T+(−92.6+77.7i)T2 1 + (-0.538 - 1.47i)T + (-92.6 + 77.7i)T^{2}
13 1+(6.82−5.72i)T+(29.3−166.i)T2 1 + (6.82 - 5.72i)T + (29.3 - 166. i)T^{2}
17 1+(16.9−9.80i)T+(144.5−250.i)T2 1 + (16.9 - 9.80i)T + (144.5 - 250. i)T^{2}
19 1+(−4.86+8.42i)T+(−180.5−312.i)T2 1 + (-4.86 + 8.42i)T + (-180.5 - 312. i)T^{2}
23 1+(13.8−2.43i)T+(497.−180.i)T2 1 + (13.8 - 2.43i)T + (497. - 180. i)T^{2}
29 1+(−7.05+8.40i)T+(−146.−828.i)T2 1 + (-7.05 + 8.40i)T + (-146. - 828. i)T^{2}
31 1+(−8.04−45.6i)T+(−903.+328.i)T2 1 + (-8.04 - 45.6i)T + (-903. + 328. i)T^{2}
37 1+(−7.89−13.6i)T+(−684.5+1.18e3i)T2 1 + (-7.89 - 13.6i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(6.59+7.86i)T+(−291.+1.65e3i)T2 1 + (6.59 + 7.86i)T + (-291. + 1.65e3i)T^{2}
43 1+(−24.6+8.96i)T+(1.41e3−1.18e3i)T2 1 + (-24.6 + 8.96i)T + (1.41e3 - 1.18e3i)T^{2}
47 1+(42.2+7.45i)T+(2.07e3+755.i)T2 1 + (42.2 + 7.45i)T + (2.07e3 + 755. i)T^{2}
53 1−41.5iT−2.80e3T2 1 - 41.5iT - 2.80e3T^{2}
59 1+(−20.6+56.8i)T+(−2.66e3−2.23e3i)T2 1 + (-20.6 + 56.8i)T + (-2.66e3 - 2.23e3i)T^{2}
61 1+(−13.2+75.3i)T+(−3.49e3−1.27e3i)T2 1 + (-13.2 + 75.3i)T + (-3.49e3 - 1.27e3i)T^{2}
67 1+(−2.09+1.75i)T+(779.−4.42e3i)T2 1 + (-2.09 + 1.75i)T + (779. - 4.42e3i)T^{2}
71 1+(−65.1+37.6i)T+(2.52e3−4.36e3i)T2 1 + (-65.1 + 37.6i)T + (2.52e3 - 4.36e3i)T^{2}
73 1+(−33.0+57.2i)T+(−2.66e3−4.61e3i)T2 1 + (-33.0 + 57.2i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(−38.8−32.5i)T+(1.08e3+6.14e3i)T2 1 + (-38.8 - 32.5i)T + (1.08e3 + 6.14e3i)T^{2}
83 1+(−21.4+25.5i)T+(−1.19e3−6.78e3i)T2 1 + (-21.4 + 25.5i)T + (-1.19e3 - 6.78e3i)T^{2}
89 1+(75.2+43.4i)T+(3.96e3+6.85e3i)T2 1 + (75.2 + 43.4i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(−132.+48.2i)T+(7.20e3−6.04e3i)T2 1 + (-132. + 48.2i)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

−15.34727899979336858470042607137, −14.02470376208640458952901207506, −12.89269698857207770685129155601, −12.12121394989362781938603140809, −9.638380765067485238328347486290, −8.980459455829226804845130196739, −8.276768219358936406297735997951, −6.39669096328354072461246350859, −4.92466492004446814686200434021, −2.01037972240035355407577531342, 2.60090867671788618794889814631, 3.93046818787463394335535337215, 6.84862986048166972930601249553, 7.78219801954984622527198079300, 9.665271963985792970321897803697, 10.26303971772962283868866827106, 11.13692501951564503005521558647, 13.25368964158125296624844262834, 13.94185056886573556779850959125, 14.84968456408918991132237872304

Graph of the ZZ-function along the critical line