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Properties

Label 2-2700-5.4-c1-0-21
Degree 22
Conductor 27002700
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 21.559621.5596
Root an. cond. 4.643234.64323
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 + 2i·13-s − 3i·17-s − 5·19-s − 3i·23-s − 6·29-s + 5·31-s − 2i·37-s − 12·41-s + 8i·43-s − 12i·47-s + 3·49-s + 3i·53-s + 6·59-s − 7·61-s + ⋯
L(s)  = 1  − 0.755i·7-s + 0.554i·13-s − 0.727i·17-s − 1.14·19-s − 0.625i·23-s − 1.11·29-s + 0.898·31-s − 0.328i·37-s − 1.87·41-s + 1.21i·43-s − 1.75i·47-s + 0.428·49-s + 0.412i·53-s + 0.781·59-s − 0.896·61-s + ⋯

Functional equation

Λ(s)=(2700s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2700s/2 ΓC(s+1/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27002700    =    22⋅33⋅522^{2} \cdot 3^{3} \cdot 5^{2}
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 21.559621.5596
Root analytic conductor: 4.643234.64323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2700(649,⋅)\chi_{2700} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2700, ( :1/2), −0.894+0.447i)(2,\ 2700,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 0.62987868560.6298786856
L(12)L(\frac12) ≈\approx 0.62987868560.6298786856
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+11T2 1 + 11T^{2}
13 1−2iT−13T2 1 - 2iT - 13T^{2}
17 1+3iT−17T2 1 + 3iT - 17T^{2}
19 1+5T+19T2 1 + 5T + 19T^{2}
23 1+3iT−23T2 1 + 3iT - 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1−5T+31T2 1 - 5T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1+12T+41T2 1 + 12T + 41T^{2}
43 1−8iT−43T2 1 - 8iT - 43T^{2}
47 1+12iT−47T2 1 + 12iT - 47T^{2}
53 1−3iT−53T2 1 - 3iT - 53T^{2}
59 1−6T+59T2 1 - 6T + 59T^{2}
61 1+7T+61T2 1 + 7T + 61T^{2}
67 1+2iT−67T2 1 + 2iT - 67T^{2}
71 1+12T+71T2 1 + 12T + 71T^{2}
73 1+16iT−73T2 1 + 16iT - 73T^{2}
79 1−T+79T2 1 - T + 79T^{2}
83 1−15iT−83T2 1 - 15iT - 83T^{2}
89 1+12T+89T2 1 + 12T + 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

−8.533760035414045311405967542191, −7.74070404352300971470179119877, −6.90644493267818336761274269203, −6.43073832395284345494871849228, −5.32564962376082343222952214163, −4.47530553601358451172067260392, −3.81462926068523236908028476419, −2.69345992915804009536340163311, −1.61374302097108356936335356159, −0.19522241933984479245536358872, 1.54933301357522248129436025119, 2.53406835853089939199932237955, 3.51657820152373293147002787362, 4.43171207048348375439391797315, 5.43131183881456093268905833899, 5.99494501538510585038114112460, 6.83193214428123948603486036627, 7.74900217606276847822930402353, 8.507701003224320979849503734115, 8.965139625393844551446875413928

Graph of the ZZ-function along the critical line