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Properties

Label 2-20e2-5.4-c3-0-17
Degree 22
Conductor 400400
Sign 0.894+0.447i0.894 + 0.447i
Analytic cond. 23.600723.6007
Root an. cond. 4.858064.85806
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5i·3-s − 2i·7-s + 2·9-s − 39·11-s − 84i·13-s − 61i·17-s + 151·19-s + 10·21-s − 58i·23-s + 145i·27-s − 192·29-s + 18·31-s − 195i·33-s − 138i·37-s + 420·39-s + ⋯
L(s)  = 1  + 0.962i·3-s − 0.107i·7-s + 0.0740·9-s − 1.06·11-s − 1.79i·13-s − 0.870i·17-s + 1.82·19-s + 0.103·21-s − 0.525i·23-s + 1.03i·27-s − 1.22·29-s + 0.104·31-s − 1.02i·33-s − 0.613i·37-s + 1.72·39-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+3/2) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 0.894+0.447i0.894 + 0.447i
Analytic conductor: 23.600723.6007
Root analytic conductor: 4.858064.85806
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ400(49,⋅)\chi_{400} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :3/2), 0.894+0.447i)(2,\ 400,\ (\ :3/2),\ 0.894 + 0.447i)

Particular Values

L(2)L(2) ≈\approx 1.6416478251.641647825
L(12)L(\frac12) ≈\approx 1.6416478251.641647825
L(52)L(\frac{5}{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
good3 1−5iT−27T2 1 - 5iT - 27T^{2}
7 1+2iT−343T2 1 + 2iT - 343T^{2}
11 1+39T+1.33e3T2 1 + 39T + 1.33e3T^{2}
13 1+84iT−2.19e3T2 1 + 84iT - 2.19e3T^{2}
17 1+61iT−4.91e3T2 1 + 61iT - 4.91e3T^{2}
19 1−151T+6.85e3T2 1 - 151T + 6.85e3T^{2}
23 1+58iT−1.21e4T2 1 + 58iT - 1.21e4T^{2}
29 1+192T+2.43e4T2 1 + 192T + 2.43e4T^{2}
31 1−18T+2.97e4T2 1 - 18T + 2.97e4T^{2}
37 1+138iT−5.06e4T2 1 + 138iT - 5.06e4T^{2}
41 1−229T+6.89e4T2 1 - 229T + 6.89e4T^{2}
43 1+164iT−7.95e4T2 1 + 164iT - 7.95e4T^{2}
47 1−212iT−1.03e5T2 1 - 212iT - 1.03e5T^{2}
53 1+578iT−1.48e5T2 1 + 578iT - 1.48e5T^{2}
59 1+336T+2.05e5T2 1 + 336T + 2.05e5T^{2}
61 1−858T+2.26e5T2 1 - 858T + 2.26e5T^{2}
67 1−209iT−3.00e5T2 1 - 209iT - 3.00e5T^{2}
71 1−780T+3.57e5T2 1 - 780T + 3.57e5T^{2}
73 1−403iT−3.89e5T2 1 - 403iT - 3.89e5T^{2}
79 1+230T+4.93e5T2 1 + 230T + 4.93e5T^{2}
83 1+1.29e3iT−5.71e5T2 1 + 1.29e3iT - 5.71e5T^{2}
89 1−1.36e3T+7.04e5T2 1 - 1.36e3T + 7.04e5T^{2}
97 1−382iT−9.12e5T2 1 - 382iT - 9.12e5T^{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

−10.58128241432074999638222300701, −9.990878595612598660703277764377, −9.209445740719670952106610329777, −7.910282377596835729813208521935, −7.28530267314271904419522932127, −5.50418318458000514619121336293, −5.12173810887201299868240399204, −3.70185621237209127973616967117, −2.73829305316966324934325726136, −0.60961234309670321078894777049, 1.26339501294110320429213289339, 2.33795150742758746190031377900, 3.88474608032378130037057038908, 5.21753486656324671773540152447, 6.30567275052962288945521705611, 7.29851259318978972731056174477, 7.85133730311744519298889962255, 9.103346589423949317988264741806, 9.918054300281964762245625564133, 11.12465080072562340770867535757

Graph of the ZZ-function along the critical line