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Properties

Label 2-5400-5.4-c1-0-20
Degree 22
Conductor 54005400
Sign −0.447−0.894i-0.447 - 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  + 4i·7-s − 2·11-s + 4i·13-s + i·17-s + 5·19-s − 5i·23-s + 8·29-s + 7·31-s + 6i·37-s − 6·41-s − 2i·43-s + 8i·47-s − 9·49-s − 9i·53-s + 4·59-s + ⋯
L(s)  = 1  + 1.51i·7-s − 0.603·11-s + 1.10i·13-s + 0.242i·17-s + 1.14·19-s − 1.04i·23-s + 1.48·29-s + 1.25·31-s + 0.986i·37-s − 0.937·41-s − 0.304i·43-s + 1.16i·47-s − 1.28·49-s − 1.23i·53-s + 0.520·59-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: −0.447−0.894i-0.447 - 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 1.6709084211.670908421
L(12)L(\frac12) ≈\approx 1.6709084211.670908421
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 1−4iT−7T2 1 - 4iT - 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1−4iT−13T2 1 - 4iT - 13T^{2}
17 1−iT−17T2 1 - iT - 17T^{2}
19 1−5T+19T2 1 - 5T + 19T^{2}
23 1+5iT−23T2 1 + 5iT - 23T^{2}
29 1−8T+29T2 1 - 8T + 29T^{2}
31 1−7T+31T2 1 - 7T + 31T^{2}
37 1−6iT−37T2 1 - 6iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1+2iT−43T2 1 + 2iT - 43T^{2}
47 1−8iT−47T2 1 - 8iT - 47T^{2}
53 1+9iT−53T2 1 + 9iT - 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1−13T+61T2 1 - 13T + 61T^{2}
67 1−10iT−67T2 1 - 10iT - 67T^{2}
71 1−6T+71T2 1 - 6T + 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1+9T+79T2 1 + 9T + 79T^{2}
83 1−17iT−83T2 1 - 17iT - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1−8iT−97T2 1 - 8iT - 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

−8.457513604577591734207474731755, −7.912628774792344586356678604113, −6.64839939934742245942584932295, −6.48498415433447582404446566068, −5.34373350150100179052004926480, −4.98451315301008518332294637764, −4.00607476183968351693958705982, −2.80163996766485035779918020571, −2.44384031563773400378362663838, −1.21879466829391126529469352461, 0.49394191554712047749289178925, 1.28163865748179823440418228252, 2.73570912604652435023129315319, 3.38695235142391045594797925299, 4.22034593240676834943524441166, 5.07645861724610998857834536149, 5.63898729066692991743344834641, 6.71930137280208014404550719121, 7.27377081650651491099368540753, 7.888779604446513700235520831823

Graph of the ZZ-function along the critical line