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Properties

Label 2-5400-5.4-c1-0-42
Degree 22
Conductor 54005400
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 43.119243.1192
Root an. cond. 6.566526.56652
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·7-s − 4·11-s − 2i·13-s + 5i·17-s + 5·19-s − i·23-s − 2·29-s + 7·31-s + 6i·37-s + 4i·43-s + 4i·47-s + 3·49-s − 9i·53-s + 14·59-s − 11·61-s + ⋯
L(s)  = 1  − 0.755i·7-s − 1.20·11-s − 0.554i·13-s + 1.21i·17-s + 1.14·19-s − 0.208i·23-s − 0.371·29-s + 1.25·31-s + 0.986i·37-s + 0.609i·43-s + 0.583i·47-s + 0.428·49-s − 1.23i·53-s + 1.82·59-s − 1.40·61-s + ⋯

Functional equation

Λ(s)=(5400s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5400s/2 ΓC(s+1/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 54005400    =    23⋅33⋅522^{3} \cdot 3^{3} \cdot 5^{2}
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 43.119243.1192
Root analytic conductor: 6.566526.56652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5400(649,⋅)\chi_{5400} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5400, ( :1/2), 0.447+0.894i)(2,\ 5400,\ (\ :1/2),\ 0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.5724303361.572430336
L(12)L(\frac12) ≈\approx 1.5724303361.572430336
L(32)L(\frac{3}{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
3 1 1
5 1 1
good7 1+2iT−7T2 1 + 2iT - 7T^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1−5iT−17T2 1 - 5iT - 17T^{2}
19 1−5T+19T2 1 - 5T + 19T^{2}
23 1+iT−23T2 1 + iT - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1−7T+31T2 1 - 7T + 31T^{2}
37 1−6iT−37T2 1 - 6iT - 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1−4iT−47T2 1 - 4iT - 47T^{2}
53 1+9iT−53T2 1 + 9iT - 53T^{2}
59 1−14T+59T2 1 - 14T + 59T^{2}
61 1+11T+61T2 1 + 11T + 61T^{2}
67 1+14iT−67T2 1 + 14iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+12iT−73T2 1 + 12iT - 73T^{2}
79 1−3T+79T2 1 - 3T + 79T^{2}
83 1−iT−83T2 1 - iT - 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+16iT−97T2 1 + 16iT - 97T^{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

−7.975715896203517819245614427972, −7.51341788346161165351386086067, −6.62069475559202915648586693719, −5.88320244115332881198355048646, −5.11400302517237359093259604316, −4.42867012941824790532438985113, −3.45089141863871636893810066595, −2.81095020440496331154088398882, −1.62541473600245785047055149123, −0.51320670619519088892062977641, 0.902818176387185737754311669214, 2.32446047100163260834764846634, 2.75859437368975164278226958323, 3.80465457221140442853936743674, 4.84453364938573235203482031350, 5.38505375582839570961371897164, 5.97811384516288082467007100774, 7.10789473179686956302803286314, 7.44615503305948131248514102896, 8.345270183361947044599939969315

Graph of the ZZ-function along the critical line