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Properties

Label 2-54-27.14-c2-0-3
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(41,⋅)\chi_{54} (41, \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

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

Graph of the ZZ-function along the critical line