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Properties

Label 2-800-40.19-c4-0-6
Degree 22
Conductor 800800
Sign 0.775−0.631i0.775 - 0.631i
Analytic cond. 82.695982.6959
Root an. cond. 9.093739.09373
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.51i·3-s − 78.3·7-s + 50.5·9-s − 123.·11-s − 229.·13-s − 557. i·17-s − 334.·19-s + 432. i·21-s + 244.·23-s − 725. i·27-s − 301. i·29-s + 339. i·31-s + 679. i·33-s − 187.·37-s + 1.26e3i·39-s + ⋯
L(s)  = 1  − 0.612i·3-s − 1.59·7-s + 0.624·9-s − 1.01·11-s − 1.35·13-s − 1.92i·17-s − 0.925·19-s + 0.979i·21-s + 0.461·23-s − 0.995i·27-s − 0.358i·29-s + 0.353i·31-s + 0.623i·33-s − 0.136·37-s + 0.833i·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.775−0.631i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+2) L(s)=((0.775−0.631i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.775−0.631i0.775 - 0.631i
Analytic conductor: 82.695982.6959
Root analytic conductor: 9.093739.09373
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ800(399,⋅)\chi_{800} (399, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :2), 0.775−0.631i)(2,\ 800,\ (\ :2),\ 0.775 - 0.631i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.52037420430.5203742043
L(12)L(\frac12) ≈\approx 0.52037420430.5203742043
L(3)L(3) 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+5.51iT−81T2 1 + 5.51iT - 81T^{2}
7 1+78.3T+2.40e3T2 1 + 78.3T + 2.40e3T^{2}
11 1+123.T+1.46e4T2 1 + 123.T + 1.46e4T^{2}
13 1+229.T+2.85e4T2 1 + 229.T + 2.85e4T^{2}
17 1+557.iT−8.35e4T2 1 + 557. iT - 8.35e4T^{2}
19 1+334.T+1.30e5T2 1 + 334.T + 1.30e5T^{2}
23 1−244.T+2.79e5T2 1 - 244.T + 2.79e5T^{2}
29 1+301.iT−7.07e5T2 1 + 301. iT - 7.07e5T^{2}
31 1−339.iT−9.23e5T2 1 - 339. iT - 9.23e5T^{2}
37 1+187.T+1.87e6T2 1 + 187.T + 1.87e6T^{2}
41 1−1.28e3T+2.82e6T2 1 - 1.28e3T + 2.82e6T^{2}
43 1−1.07e3iT−3.41e6T2 1 - 1.07e3iT - 3.41e6T^{2}
47 1+3.24e3T+4.87e6T2 1 + 3.24e3T + 4.87e6T^{2}
53 1−2.02e3T+7.89e6T2 1 - 2.02e3T + 7.89e6T^{2}
59 1−555.T+1.21e7T2 1 - 555.T + 1.21e7T^{2}
61 1−3.18e3iT−1.38e7T2 1 - 3.18e3iT - 1.38e7T^{2}
67 1−3.43e3iT−2.01e7T2 1 - 3.43e3iT - 2.01e7T^{2}
71 1−2.40e3iT−2.54e7T2 1 - 2.40e3iT - 2.54e7T^{2}
73 1−2.09e3iT−2.83e7T2 1 - 2.09e3iT - 2.83e7T^{2}
79 1−2.93e3iT−3.89e7T2 1 - 2.93e3iT - 3.89e7T^{2}
83 1+8.32e3iT−4.74e7T2 1 + 8.32e3iT - 4.74e7T^{2}
89 1−406.T+6.27e7T2 1 - 406.T + 6.27e7T^{2}
97 1−1.76e4iT−8.85e7T2 1 - 1.76e4iT - 8.85e7T^{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

−9.836417574925723879049887923168, −9.123474446516929864670318170169, −7.80720187751612523385196696739, −7.10362638612391563630577190678, −6.59286934115549429079234114463, −5.38312961163151162114270689754, −4.43167616778088620790112964280, −2.96796671133221929814344285677, −2.38291132901383866504246349141, −0.62927297853766497970783662555, 0.18338564345894239837767431180, 2.05776900879149324239211520270, 3.18891279050029484793951743895, 4.07366436288220159968754680414, 5.05895267911823616560084834703, 6.13622978968147202680424011778, 6.92351106188002632007933115995, 7.87368492802795559430025558445, 8.951744439821716253880817342554, 9.821587258753542143956125951188

Graph of the ZZ-function along the critical line