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Properties

Label 2-54-27.2-c2-0-4
Degree 22
Conductor 5454
Sign 0.886+0.462i0.886 + 0.462i
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 + (−0.101 − 2.99i)3-s + (1.87 + 0.684i)4-s + (0.379 + 0.451i)5-s + (0.595 − 4.20i)6-s + (2.96 − 1.07i)7-s + (2.44 + 1.41i)8-s + (−8.97 + 0.607i)9-s + (0.416 + 0.722i)10-s + (−6.03 + 7.19i)11-s + (1.86 − 5.70i)12-s + (2.11 + 12.0i)13-s + (4.39 − 0.774i)14-s + (1.31 − 1.18i)15-s + (3.06 + 2.57i)16-s + (−24.5 + 14.1i)17-s + ⋯
L(s)  = 1  + (0.696 + 0.122i)2-s + (−0.0337 − 0.999i)3-s + (0.469 + 0.171i)4-s + (0.0758 + 0.0903i)5-s + (0.0992 − 0.700i)6-s + (0.423 − 0.154i)7-s + (0.306 + 0.176i)8-s + (−0.997 + 0.0674i)9-s + (0.0416 + 0.0722i)10-s + (−0.548 + 0.653i)11-s + (0.155 − 0.475i)12-s + (0.163 + 0.924i)13-s + (0.313 − 0.0553i)14-s + (0.0877 − 0.0788i)15-s + (0.191 + 0.160i)16-s + (−1.44 + 0.832i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.886+0.462i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((0.886+0.462i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.886+0.462i0.886 + 0.462i
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), 0.886+0.462i)(2,\ 54,\ (\ :1),\ 0.886 + 0.462i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.56757−0.384040i1.56757 - 0.384040i
L(12)L(\frac12) ≈\approx 1.56757−0.384040i1.56757 - 0.384040i
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+(0.101+2.99i)T 1 + (0.101 + 2.99i)T
good5 1+(−0.379−0.451i)T+(−4.34+24.6i)T2 1 + (-0.379 - 0.451i)T + (-4.34 + 24.6i)T^{2}
7 1+(−2.96+1.07i)T+(37.5−31.4i)T2 1 + (-2.96 + 1.07i)T + (37.5 - 31.4i)T^{2}
11 1+(6.03−7.19i)T+(−21.0−119.i)T2 1 + (6.03 - 7.19i)T + (-21.0 - 119. i)T^{2}
13 1+(−2.11−12.0i)T+(−158.+57.8i)T2 1 + (-2.11 - 12.0i)T + (-158. + 57.8i)T^{2}
17 1+(24.5−14.1i)T+(144.5−250.i)T2 1 + (24.5 - 14.1i)T + (144.5 - 250. i)T^{2}
19 1+(−11.2+19.4i)T+(−180.5−312.i)T2 1 + (-11.2 + 19.4i)T + (-180.5 - 312. i)T^{2}
23 1+(−3.44+9.46i)T+(−405.−340.i)T2 1 + (-3.44 + 9.46i)T + (-405. - 340. i)T^{2}
29 1+(23.6+4.17i)T+(790.+287.i)T2 1 + (23.6 + 4.17i)T + (790. + 287. i)T^{2}
31 1+(−42.7−15.5i)T+(736.+617.i)T2 1 + (-42.7 - 15.5i)T + (736. + 617. i)T^{2}
37 1+(15.7+27.2i)T+(−684.5+1.18e3i)T2 1 + (15.7 + 27.2i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(−69.7+12.3i)T+(1.57e3−574.i)T2 1 + (-69.7 + 12.3i)T + (1.57e3 - 574. i)T^{2}
43 1+(11.1+9.36i)T+(321.+1.82e3i)T2 1 + (11.1 + 9.36i)T + (321. + 1.82e3i)T^{2}
47 1+(18.9+51.9i)T+(−1.69e3+1.41e3i)T2 1 + (18.9 + 51.9i)T + (-1.69e3 + 1.41e3i)T^{2}
53 1+25.4iT−2.80e3T2 1 + 25.4iT - 2.80e3T^{2}
59 1+(−18.4−21.9i)T+(−604.+3.42e3i)T2 1 + (-18.4 - 21.9i)T + (-604. + 3.42e3i)T^{2}
61 1+(106.−38.6i)T+(2.85e3−2.39e3i)T2 1 + (106. - 38.6i)T + (2.85e3 - 2.39e3i)T^{2}
67 1+(−8.68−49.2i)T+(−4.21e3+1.53e3i)T2 1 + (-8.68 - 49.2i)T + (-4.21e3 + 1.53e3i)T^{2}
71 1+(7.59−4.38i)T+(2.52e3−4.36e3i)T2 1 + (7.59 - 4.38i)T + (2.52e3 - 4.36e3i)T^{2}
73 1+(11.7−20.3i)T+(−2.66e3−4.61e3i)T2 1 + (11.7 - 20.3i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(−23.0+130.i)T+(−5.86e3−2.13e3i)T2 1 + (-23.0 + 130. i)T + (-5.86e3 - 2.13e3i)T^{2}
83 1+(66.1+11.6i)T+(6.47e3+2.35e3i)T2 1 + (66.1 + 11.6i)T + (6.47e3 + 2.35e3i)T^{2}
89 1+(−62.9−36.3i)T+(3.96e3+6.85e3i)T2 1 + (-62.9 - 36.3i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(−120.−101.i)T+(1.63e3+9.26e3i)T2 1 + (-120. - 101. i)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

−14.77380248530032345712245542170, −13.75439647779181284400246889140, −12.94963339391984396789222045863, −11.82081126909820197230120710805, −10.79325377822480876979654928475, −8.749222102230116489596584802193, −7.34770271232115691812181668863, −6.33715799857682233554290907194, −4.62606956103435781375452131127, −2.25567298128436966450156265618, 3.11067959533681590467984962079, 4.78632062244075931598911280742, 5.87461768423812367559929941651, 7.989676169539213054501500528586, 9.460608528645908265153848791031, 10.80538916807334349239540306122, 11.54981274956734410178443121034, 13.11959561308286693741317234771, 14.12426381416141363567509675628, 15.31054401684461119467939147569

Graph of the ZZ-function along the critical line