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Properties

Label 2-800-8.5-c5-0-37
Degree 22
Conductor 800800
Sign 0.00874−0.999i0.00874 - 0.999i
Analytic cond. 128.307128.307
Root an. cond. 11.327211.3272
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.7i·3-s + 198.·7-s + 127.·9-s − 85.9i·11-s − 407. i·13-s − 1.20e3·17-s + 206. i·19-s + 2.13e3i·21-s − 2.59e3·23-s + 3.98e3i·27-s + 6.19e3i·29-s + 1.86e3·31-s + 923.·33-s + 1.47e4i·37-s + 4.37e3·39-s + ⋯
L(s)  = 1  + 0.689i·3-s + 1.53·7-s + 0.524·9-s − 0.214i·11-s − 0.668i·13-s − 1.01·17-s + 0.130i·19-s + 1.05i·21-s − 1.02·23-s + 1.05i·27-s + 1.36i·29-s + 0.348·31-s + 0.147·33-s + 1.76i·37-s + 0.460·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.00874−0.999i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00874 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+5/2) L(s)=((0.00874−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.00874 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.00874−0.999i0.00874 - 0.999i
Analytic conductor: 128.307128.307
Root analytic conductor: 11.327211.3272
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ800(401,⋅)\chi_{800} (401, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :5/2), 0.00874−0.999i)(2,\ 800,\ (\ :5/2),\ 0.00874 - 0.999i)

Particular Values

L(3)L(3) ≈\approx 2.5896106222.589610622
L(12)L(\frac12) ≈\approx 2.5896106222.589610622
L(72)L(\frac{7}{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 1−10.7iT−243T2 1 - 10.7iT - 243T^{2}
7 1−198.T+1.68e4T2 1 - 198.T + 1.68e4T^{2}
11 1+85.9iT−1.61e5T2 1 + 85.9iT - 1.61e5T^{2}
13 1+407.iT−3.71e5T2 1 + 407. iT - 3.71e5T^{2}
17 1+1.20e3T+1.41e6T2 1 + 1.20e3T + 1.41e6T^{2}
19 1−206.iT−2.47e6T2 1 - 206. iT - 2.47e6T^{2}
23 1+2.59e3T+6.43e6T2 1 + 2.59e3T + 6.43e6T^{2}
29 1−6.19e3iT−2.05e7T2 1 - 6.19e3iT - 2.05e7T^{2}
31 1−1.86e3T+2.86e7T2 1 - 1.86e3T + 2.86e7T^{2}
37 1−1.47e4iT−6.93e7T2 1 - 1.47e4iT - 6.93e7T^{2}
41 1−1.80e4T+1.15e8T2 1 - 1.80e4T + 1.15e8T^{2}
43 1+9.26e3iT−1.47e8T2 1 + 9.26e3iT - 1.47e8T^{2}
47 1+2.43e4T+2.29e8T2 1 + 2.43e4T + 2.29e8T^{2}
53 1−1.27e4iT−4.18e8T2 1 - 1.27e4iT - 4.18e8T^{2}
59 1−2.07e4iT−7.14e8T2 1 - 2.07e4iT - 7.14e8T^{2}
61 1−1.13e4iT−8.44e8T2 1 - 1.13e4iT - 8.44e8T^{2}
67 1+6.26e4iT−1.35e9T2 1 + 6.26e4iT - 1.35e9T^{2}
71 1−6.12e4T+1.80e9T2 1 - 6.12e4T + 1.80e9T^{2}
73 1+2.32e4T+2.07e9T2 1 + 2.32e4T + 2.07e9T^{2}
79 1+2.91e4T+3.07e9T2 1 + 2.91e4T + 3.07e9T^{2}
83 1−4.80e4iT−3.93e9T2 1 - 4.80e4iT - 3.93e9T^{2}
89 1−3.01e4T+5.58e9T2 1 - 3.01e4T + 5.58e9T^{2}
97 1−1.13e5T+8.58e9T2 1 - 1.13e5T + 8.58e9T^{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

−9.829055760924839840032225536085, −8.805998878001819002726515998249, −8.126612767766780811784137957575, −7.30192435106135733962752127582, −6.12846804399562366202695169181, −4.96777627182504535229515008111, −4.55062728153249575166220465507, −3.45505196578484690002281938467, −2.07626578319971181913713868359, −1.07930580398720528151792794064, 0.55134763466201568421025734177, 1.77897242583659381088160207347, 2.22312843097492048754441166216, 4.15293269760839087867100600976, 4.61901109420325951015323905405, 5.89794878421527816848210334329, 6.80634066049668857049592518242, 7.68988533834901448899621329340, 8.194924577800514045424429423428, 9.237843633260651198688089136736

Graph of the ZZ-function along the critical line