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Properties

Label 2-800-5.4-c3-0-17
Degree 22
Conductor 800800
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 47.201547.2015
Root an. cond. 6.870336.87033
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 + 10i·7-s + 2·9-s + 15·11-s + 8i·13-s + 21i·17-s + 105·19-s − 50·21-s − 10i·23-s + 145i·27-s + 20·29-s + 230·31-s + 75i·33-s + 54i·37-s − 40·39-s + ⋯
L(s)  = 1  + 0.962i·3-s + 0.539i·7-s + 0.0740·9-s + 0.411·11-s + 0.170i·13-s + 0.299i·17-s + 1.26·19-s − 0.519·21-s − 0.0906i·23-s + 1.03i·27-s + 0.128·29-s + 1.33·31-s + 0.395i·33-s + 0.239i·37-s − 0.164·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+3/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 47.201547.2015
Root analytic conductor: 6.870336.87033
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ800(449,⋅)\chi_{800} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :3/2), −0.447−0.894i)(2,\ 800,\ (\ :3/2),\ -0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 2.1304462412.130446241
L(12)L(\frac12) ≈\approx 2.1304462412.130446241
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 1−10iT−343T2 1 - 10iT - 343T^{2}
11 1−15T+1.33e3T2 1 - 15T + 1.33e3T^{2}
13 1−8iT−2.19e3T2 1 - 8iT - 2.19e3T^{2}
17 1−21iT−4.91e3T2 1 - 21iT - 4.91e3T^{2}
19 1−105T+6.85e3T2 1 - 105T + 6.85e3T^{2}
23 1+10iT−1.21e4T2 1 + 10iT - 1.21e4T^{2}
29 1−20T+2.43e4T2 1 - 20T + 2.43e4T^{2}
31 1−230T+2.97e4T2 1 - 230T + 2.97e4T^{2}
37 1−54iT−5.06e4T2 1 - 54iT - 5.06e4T^{2}
41 1+195T+6.89e4T2 1 + 195T + 6.89e4T^{2}
43 1+300iT−7.95e4T2 1 + 300iT - 7.95e4T^{2}
47 1−480iT−1.03e5T2 1 - 480iT - 1.03e5T^{2}
53 1−322iT−1.48e5T2 1 - 322iT - 1.48e5T^{2}
59 1−560T+2.05e5T2 1 - 560T + 2.05e5T^{2}
61 1+730T+2.26e5T2 1 + 730T + 2.26e5T^{2}
67 1+255iT−3.00e5T2 1 + 255iT - 3.00e5T^{2}
71 1−40T+3.57e5T2 1 - 40T + 3.57e5T^{2}
73 1−317iT−3.89e5T2 1 - 317iT - 3.89e5T^{2}
79 1+830T+4.93e5T2 1 + 830T + 4.93e5T^{2}
83 1−75iT−5.71e5T2 1 - 75iT - 5.71e5T^{2}
89 1−705T+7.04e5T2 1 - 705T + 7.04e5T^{2}
97 1−1.43e3iT−9.12e5T2 1 - 1.43e3iT - 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

−10.05933523694180690769316812377, −9.390462472179069596155106907535, −8.695014365901603831648621809636, −7.63561133962991080667685581651, −6.60857381789066001051997728776, −5.58153553336405977469609779422, −4.71570131924286179387925384088, −3.82731252784844201825549473418, −2.78126844935986425972026293445, −1.26488385555692970705924787355, 0.64613833073178231131757435988, 1.54790423162024687556825623203, 2.88791181688403196196396084694, 4.07163811184688328759868427161, 5.18653088557902217491367584193, 6.33301783298817688197392994370, 7.06974186968484141758342007936, 7.70072005272500346146200503196, 8.588597613853944861855092057133, 9.716079964802622421143196220288

Graph of the ZZ-function along the critical line