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Properties

Label 2-2800-5.4-c1-0-39
Degree 22
Conductor 28002800
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 22.358122.3581
Root an. cond. 4.728434.72843
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  + 3.23i·3-s + i·7-s − 7.47·9-s + 0.236·11-s + 1.23i·13-s − 2.47i·17-s − 4.47·19-s − 3.23·21-s − 6.23i·23-s − 14.4i·27-s − 5·29-s − 3.70·31-s + 0.763i·33-s − 3i·37-s − 4.00·39-s + ⋯
L(s)  = 1  + 1.86i·3-s + 0.377i·7-s − 2.49·9-s + 0.0711·11-s + 0.342i·13-s − 0.599i·17-s − 1.02·19-s − 0.706·21-s − 1.30i·23-s − 2.78i·27-s − 0.928·29-s − 0.666·31-s + 0.132i·33-s − 0.493i·37-s − 0.640·39-s + ⋯

Functional equation

Λ(s)=(2800s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2800s/2 ΓC(s+1/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{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: 28002800    =    24⋅52⋅72^{4} \cdot 5^{2} \cdot 7
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 22.358122.3581
Root analytic conductor: 4.728434.72843
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2800(449,⋅)\chi_{2800} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2800, ( :1/2), 0.447+0.894i)(2,\ 2800,\ (\ :1/2),\ 0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.13977184650.1397718465
L(12)L(\frac12) ≈\approx 0.13977184650.1397718465
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
5 1 1
7 1−iT 1 - iT
good3 1−3.23iT−3T2 1 - 3.23iT - 3T^{2}
11 1−0.236T+11T2 1 - 0.236T + 11T^{2}
13 1−1.23iT−13T2 1 - 1.23iT - 13T^{2}
17 1+2.47iT−17T2 1 + 2.47iT - 17T^{2}
19 1+4.47T+19T2 1 + 4.47T + 19T^{2}
23 1+6.23iT−23T2 1 + 6.23iT - 23T^{2}
29 1+5T+29T2 1 + 5T + 29T^{2}
31 1+3.70T+31T2 1 + 3.70T + 31T^{2}
37 1+3iT−37T2 1 + 3iT - 37T^{2}
41 1−4.76T+41T2 1 - 4.76T + 41T^{2}
43 1+1.76iT−43T2 1 + 1.76iT - 43T^{2}
47 1+2iT−47T2 1 + 2iT - 47T^{2}
53 1−8.47iT−53T2 1 - 8.47iT - 53T^{2}
59 1−11.7T+59T2 1 - 11.7T + 59T^{2}
61 1+9.70T+61T2 1 + 9.70T + 61T^{2}
67 1+4.23iT−67T2 1 + 4.23iT - 67T^{2}
71 1+8.70T+71T2 1 + 8.70T + 71T^{2}
73 1+8.76iT−73T2 1 + 8.76iT - 73T^{2}
79 1+11.1T+79T2 1 + 11.1T + 79T^{2}
83 1−7.70iT−83T2 1 - 7.70iT - 83T^{2}
89 1+17.2T+89T2 1 + 17.2T + 89T^{2}
97 1+5.23iT−97T2 1 + 5.23iT - 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

−8.921952498810517910930307344427, −8.283541047884558335481791244563, −7.13362542137320029036728186546, −6.05616830021155433746266937816, −5.47942157351040370949303088165, −4.52485772829227384204003152822, −4.13144706755480360444829486602, −3.11771625041123512885857908301, −2.24949752254809086456957177626, −0.04413071265307839223630961030, 1.27178433283752652936136872820, 2.00832736100600395018865565229, 3.04953882838238640811421960804, 4.06221795813139764516577667612, 5.45920589556857131676084899165, 5.98479716112337461782433213735, 6.82876511315749359168968123401, 7.36389882689012113726433269499, 8.036929703147324235629660299509, 8.616607872958680640775885536164

Graph of the ZZ-function along the critical line