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Properties

Label 2-54-27.5-c2-0-4
Degree 22
Conductor 5454
Sign 0.959−0.281i0.959 - 0.281i
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 + (2.14 − 2.09i)3-s + (−0.347 + 1.96i)4-s + (0.387 + 1.06i)5-s + (4.22 + 0.418i)6-s + (−0.332 − 1.88i)7-s + (−2.44 + 1.41i)8-s + (0.205 − 8.99i)9-s + (−0.800 + 1.38i)10-s + (−6.05 + 16.6i)11-s + (3.38 + 4.95i)12-s + (−9.14 − 7.67i)13-s + (1.73 − 2.07i)14-s + (3.06 + 1.47i)15-s + (−3.75 − 1.36i)16-s + (−2.93 − 1.69i)17-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (0.715 − 0.698i)3-s + (−0.0868 + 0.492i)4-s + (0.0774 + 0.212i)5-s + (0.703 + 0.0696i)6-s + (−0.0474 − 0.269i)7-s + (−0.306 + 0.176i)8-s + (0.0228 − 0.999i)9-s + (−0.0800 + 0.138i)10-s + (−0.550 + 1.51i)11-s + (0.282 + 0.412i)12-s + (−0.703 − 0.590i)13-s + (0.124 − 0.148i)14-s + (0.204 + 0.0980i)15-s + (−0.234 − 0.0855i)16-s + (−0.172 − 0.0998i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.959−0.281i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((0.959−0.281i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.959−0.281i0.959 - 0.281i
Analytic conductor: 1.471391.47139
Root analytic conductor: 1.213011.21301
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ54(5,⋅)\chi_{54} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1), 0.959−0.281i)(2,\ 54,\ (\ :1),\ 0.959 - 0.281i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.61130+0.231750i1.61130 + 0.231750i
L(12)L(\frac12) ≈\approx 1.61130+0.231750i1.61130 + 0.231750i
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+(−2.14+2.09i)T 1 + (-2.14 + 2.09i)T
good5 1+(−0.387−1.06i)T+(−19.1+16.0i)T2 1 + (-0.387 - 1.06i)T + (-19.1 + 16.0i)T^{2}
7 1+(0.332+1.88i)T+(−46.0+16.7i)T2 1 + (0.332 + 1.88i)T + (-46.0 + 16.7i)T^{2}
11 1+(6.05−16.6i)T+(−92.6−77.7i)T2 1 + (6.05 - 16.6i)T + (-92.6 - 77.7i)T^{2}
13 1+(9.14+7.67i)T+(29.3+166.i)T2 1 + (9.14 + 7.67i)T + (29.3 + 166. i)T^{2}
17 1+(2.93+1.69i)T+(144.5+250.i)T2 1 + (2.93 + 1.69i)T + (144.5 + 250. i)T^{2}
19 1+(10.2+17.7i)T+(−180.5+312.i)T2 1 + (10.2 + 17.7i)T + (-180.5 + 312. i)T^{2}
23 1+(−0.678−0.119i)T+(497.+180.i)T2 1 + (-0.678 - 0.119i)T + (497. + 180. i)T^{2}
29 1+(−34.6−41.3i)T+(−146.+828.i)T2 1 + (-34.6 - 41.3i)T + (-146. + 828. i)T^{2}
31 1+(7.33−41.6i)T+(−903.−328.i)T2 1 + (7.33 - 41.6i)T + (-903. - 328. i)T^{2}
37 1+(−8.24+14.2i)T+(−684.5−1.18e3i)T2 1 + (-8.24 + 14.2i)T + (-684.5 - 1.18e3i)T^{2}
41 1+(−26.0+30.9i)T+(−291.−1.65e3i)T2 1 + (-26.0 + 30.9i)T + (-291. - 1.65e3i)T^{2}
43 1+(−66.4−24.1i)T+(1.41e3+1.18e3i)T2 1 + (-66.4 - 24.1i)T + (1.41e3 + 1.18e3i)T^{2}
47 1+(62.9−11.1i)T+(2.07e3−755.i)T2 1 + (62.9 - 11.1i)T + (2.07e3 - 755. i)T^{2}
53 1+17.6iT−2.80e3T2 1 + 17.6iT - 2.80e3T^{2}
59 1+(19.1+52.5i)T+(−2.66e3+2.23e3i)T2 1 + (19.1 + 52.5i)T + (-2.66e3 + 2.23e3i)T^{2}
61 1+(−11.5−65.3i)T+(−3.49e3+1.27e3i)T2 1 + (-11.5 - 65.3i)T + (-3.49e3 + 1.27e3i)T^{2}
67 1+(17.2+14.5i)T+(779.+4.42e3i)T2 1 + (17.2 + 14.5i)T + (779. + 4.42e3i)T^{2}
71 1+(43.0+24.8i)T+(2.52e3+4.36e3i)T2 1 + (43.0 + 24.8i)T + (2.52e3 + 4.36e3i)T^{2}
73 1+(45.2+78.3i)T+(−2.66e3+4.61e3i)T2 1 + (45.2 + 78.3i)T + (-2.66e3 + 4.61e3i)T^{2}
79 1+(57.2−48.0i)T+(1.08e3−6.14e3i)T2 1 + (57.2 - 48.0i)T + (1.08e3 - 6.14e3i)T^{2}
83 1+(−8.15−9.72i)T+(−1.19e3+6.78e3i)T2 1 + (-8.15 - 9.72i)T + (-1.19e3 + 6.78e3i)T^{2}
89 1+(−58.5+33.7i)T+(3.96e3−6.85e3i)T2 1 + (-58.5 + 33.7i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(44.2+16.1i)T+(7.20e3+6.04e3i)T2 1 + (44.2 + 16.1i)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

−14.90772595066502435647568610329, −14.22230990021033691002408344258, −12.88226591393590831507570933532, −12.39250973225327104725136708672, −10.41008307445508280843543008784, −8.926589304882862890056795410043, −7.51851863961740118387786042875, −6.76710677094160569160493727486, −4.75726040545208337705403596433, −2.69634484913065404676123428739, 2.68933398913361649537474341982, 4.28662529850780581039586903138, 5.81215819023315919655327727023, 8.103107543721627101831930090124, 9.258061038737906162539983676691, 10.42199727332834172671937324208, 11.53409530170406675839759242730, 13.00294870510459137150550750266, 13.93630217019487460819943914325, 14.87122689168945020002994630760

Graph of the ZZ-function along the critical line