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Properties

Label 2-54-27.2-c2-0-3
Degree 22
Conductor 5454
Sign 0.980−0.195i0.980 - 0.195i
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 + (2.89 + 0.783i)3-s + (1.87 + 0.684i)4-s + (−3.55 − 4.24i)5-s + (3.84 + 1.80i)6-s + (−10.2 + 3.73i)7-s + (2.44 + 1.41i)8-s + (7.77 + 4.53i)9-s + (−3.91 − 6.77i)10-s + (2.64 − 3.15i)11-s + (4.90 + 3.45i)12-s + (−0.953 − 5.40i)13-s + (−15.2 + 2.68i)14-s + (−6.98 − 15.0i)15-s + (3.06 + 2.57i)16-s + (4.10 − 2.37i)17-s + ⋯
L(s)  = 1  + (0.696 + 0.122i)2-s + (0.965 + 0.261i)3-s + (0.469 + 0.171i)4-s + (−0.711 − 0.848i)5-s + (0.640 + 0.300i)6-s + (−1.46 + 0.533i)7-s + (0.306 + 0.176i)8-s + (0.863 + 0.504i)9-s + (−0.391 − 0.677i)10-s + (0.240 − 0.286i)11-s + (0.408 + 0.287i)12-s + (−0.0733 − 0.416i)13-s + (−1.08 + 0.191i)14-s + (−0.465 − 1.00i)15-s + (0.191 + 0.160i)16-s + (0.241 − 0.139i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.980−0.195i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((0.980−0.195i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.980−0.195i0.980 - 0.195i
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), 0.980−0.195i)(2,\ 54,\ (\ :1),\ 0.980 - 0.195i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.78451+0.175909i1.78451 + 0.175909i
L(12)L(\frac12) ≈\approx 1.78451+0.175909i1.78451 + 0.175909i
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+(−2.89−0.783i)T 1 + (-2.89 - 0.783i)T
good5 1+(3.55+4.24i)T+(−4.34+24.6i)T2 1 + (3.55 + 4.24i)T + (-4.34 + 24.6i)T^{2}
7 1+(10.2−3.73i)T+(37.5−31.4i)T2 1 + (10.2 - 3.73i)T + (37.5 - 31.4i)T^{2}
11 1+(−2.64+3.15i)T+(−21.0−119.i)T2 1 + (-2.64 + 3.15i)T + (-21.0 - 119. i)T^{2}
13 1+(0.953+5.40i)T+(−158.+57.8i)T2 1 + (0.953 + 5.40i)T + (-158. + 57.8i)T^{2}
17 1+(−4.10+2.37i)T+(144.5−250.i)T2 1 + (-4.10 + 2.37i)T + (144.5 - 250. i)T^{2}
19 1+(17.1−29.7i)T+(−180.5−312.i)T2 1 + (17.1 - 29.7i)T + (-180.5 - 312. i)T^{2}
23 1+(−12.6+34.6i)T+(−405.−340.i)T2 1 + (-12.6 + 34.6i)T + (-405. - 340. i)T^{2}
29 1+(5.18+0.914i)T+(790.+287.i)T2 1 + (5.18 + 0.914i)T + (790. + 287. i)T^{2}
31 1+(−34.9−12.7i)T+(736.+617.i)T2 1 + (-34.9 - 12.7i)T + (736. + 617. i)T^{2}
37 1+(12.1+21.1i)T+(−684.5+1.18e3i)T2 1 + (12.1 + 21.1i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(22.6−3.98i)T+(1.57e3−574.i)T2 1 + (22.6 - 3.98i)T + (1.57e3 - 574. i)T^{2}
43 1+(−39.1−32.8i)T+(321.+1.82e3i)T2 1 + (-39.1 - 32.8i)T + (321. + 1.82e3i)T^{2}
47 1+(−28.3−77.8i)T+(−1.69e3+1.41e3i)T2 1 + (-28.3 - 77.8i)T + (-1.69e3 + 1.41e3i)T^{2}
53 1+16.2iT−2.80e3T2 1 + 16.2iT - 2.80e3T^{2}
59 1+(45.6+54.4i)T+(−604.+3.42e3i)T2 1 + (45.6 + 54.4i)T + (-604. + 3.42e3i)T^{2}
61 1+(74.1−26.9i)T+(2.85e3−2.39e3i)T2 1 + (74.1 - 26.9i)T + (2.85e3 - 2.39e3i)T^{2}
67 1+(12.1+68.7i)T+(−4.21e3+1.53e3i)T2 1 + (12.1 + 68.7i)T + (-4.21e3 + 1.53e3i)T^{2}
71 1+(65.9−38.0i)T+(2.52e3−4.36e3i)T2 1 + (65.9 - 38.0i)T + (2.52e3 - 4.36e3i)T^{2}
73 1+(−25.5+44.1i)T+(−2.66e3−4.61e3i)T2 1 + (-25.5 + 44.1i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(−5.51+31.2i)T+(−5.86e3−2.13e3i)T2 1 + (-5.51 + 31.2i)T + (-5.86e3 - 2.13e3i)T^{2}
83 1+(−28.7−5.06i)T+(6.47e3+2.35e3i)T2 1 + (-28.7 - 5.06i)T + (6.47e3 + 2.35e3i)T^{2}
89 1+(69.2+40.0i)T+(3.96e3+6.85e3i)T2 1 + (69.2 + 40.0i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(−36.3−30.5i)T+(1.63e3+9.26e3i)T2 1 + (-36.3 - 30.5i)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

−15.17871365251724711628717397418, −14.08222796410064882614072204960, −12.63355712919703711768016694882, −12.48076905814130625400107235185, −10.37587748409041762662043185921, −9.019511237889781641579460693233, −7.988111572277046639272926992061, −6.26239629548866079371251223349, −4.37797044776769704730544168683, −3.06368573362676867583507205708, 2.93446409796655742873161744395, 3.98932022230880128229667946263, 6.67814947063367080078474225480, 7.30357415650357885706083834594, 9.212999082114188670185167904760, 10.44874976614386010326234760048, 11.87304591500109595285912489799, 13.14160855340712958238437132175, 13.77568513012028554355616473304, 15.20216216224770645491969665634

Graph of the ZZ-function along the critical line