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Properties

Label 2-800-40.29-c3-0-4
Degree $2$
Conductor $800$
Sign $-0.734 - 0.678i$
Analytic cond. $47.2015$
Root an. cond. $6.87033$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.29·3-s − 8i·7-s + 1.00·9-s + 15.8i·11-s − 52.9·13-s + 14i·17-s − 37.0i·19-s − 42.3i·21-s + 152i·23-s − 137.·27-s + 158. i·29-s − 224·31-s + 84.0i·33-s + 243.·37-s − 280·39-s + ⋯
L(s)  = 1  + 1.01·3-s − 0.431i·7-s + 0.0370·9-s + 0.435i·11-s − 1.12·13-s + 0.199i·17-s − 0.447i·19-s − 0.439i·21-s + 1.37i·23-s − 0.980·27-s + 1.01i·29-s − 1.29·31-s + 0.443i·33-s + 1.08·37-s − 1.14·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.734 - 0.678i$
Analytic conductor: \(47.2015\)
Root analytic conductor: \(6.87033\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :3/2),\ -0.734 - 0.678i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9496272166\)
\(L(\frac12)\) \(\approx\) \(0.9496272166\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 5.29T + 27T^{2} \)
7 \( 1 + 8iT - 343T^{2} \)
11 \( 1 - 15.8iT - 1.33e3T^{2} \)
13 \( 1 + 52.9T + 2.19e3T^{2} \)
17 \( 1 - 14iT - 4.91e3T^{2} \)
19 \( 1 + 37.0iT - 6.85e3T^{2} \)
23 \( 1 - 152iT - 1.21e4T^{2} \)
29 \( 1 - 158. iT - 2.43e4T^{2} \)
31 \( 1 + 224T + 2.97e4T^{2} \)
37 \( 1 - 243.T + 5.06e4T^{2} \)
41 \( 1 + 70T + 6.89e4T^{2} \)
43 \( 1 + 439.T + 7.95e4T^{2} \)
47 \( 1 - 336iT - 1.03e5T^{2} \)
53 \( 1 + 31.7T + 1.48e5T^{2} \)
59 \( 1 - 534. iT - 2.05e5T^{2} \)
61 \( 1 - 95.2iT - 2.26e5T^{2} \)
67 \( 1 + 174.T + 3.00e5T^{2} \)
71 \( 1 - 72T + 3.57e5T^{2} \)
73 \( 1 + 294iT - 3.89e5T^{2} \)
79 \( 1 + 464T + 4.93e5T^{2} \)
83 \( 1 + 545.T + 5.71e5T^{2} \)
89 \( 1 + 266T + 7.04e5T^{2} \)
97 \( 1 + 994iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.936511745836110773478552280502, −9.356531514583465689660062336812, −8.560408759262615670784819746433, −7.51692225948040233354951466079, −7.16607552221921990541346642477, −5.73083116678656916035407862032, −4.71083258258858866598364372250, −3.61984291338591926458651502044, −2.71388434960028552481631792950, −1.60425112138716005833359310742, 0.19749737365243462584561029094, 2.06453418984421874801960520088, 2.80091466844116728181909144656, 3.85581502070439342016739702332, 5.03162542159618158424281366705, 6.02206083705438186429303512542, 7.12447500476795847599562724242, 8.054766369998452507362854110882, 8.622005786734525383311999479649, 9.472019794690649143179166188418

Graph of the $Z$-function along the critical line