↑

Properties

Label 2-800-5.4-c5-0-57
Degree $2$
Conductor $800$
Sign $0.894 + 0.447i$
Analytic cond. $128.307$
Root an. cond. $11.3272$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 18.4i·3-s + 121. i·7-s − 98.1·9-s − 438.·11-s + 758. i·13-s − 1.53e3i·17-s − 75.8·19-s − 2.25e3·21-s − 3.69e3i·23-s + 2.67e3i·27-s − 6.32e3·29-s + 2.69e3·31-s − 8.09e3i·33-s − 7.25e3i·37-s − 1.40e4·39-s + ⋯
L(s)  = 1  + 1.18i·3-s + 0.940i·7-s − 0.404·9-s − 1.09·11-s + 1.24i·13-s − 1.28i·17-s − 0.0482·19-s − 1.11·21-s − 1.45i·23-s + 0.706i·27-s − 1.39·29-s + 0.502·31-s − 1.29i·33-s − 0.870i·37-s − 1.47·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(128.307\)
Root analytic conductor: \(11.3272\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8590852244\)
\(L(\frac12)\) \(\approx\) \(0.8590852244\)
\(L(\frac{7}{2})\) 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 \( 1 - 18.4iT - 243T^{2} \)
7 \( 1 - 121. iT - 1.68e4T^{2} \)
11 \( 1 + 438.T + 1.61e5T^{2} \)
13 \( 1 - 758. iT - 3.71e5T^{2} \)
17 \( 1 + 1.53e3iT - 1.41e6T^{2} \)
19 \( 1 + 75.8T + 2.47e6T^{2} \)
23 \( 1 + 3.69e3iT - 6.43e6T^{2} \)
29 \( 1 + 6.32e3T + 2.05e7T^{2} \)
31 \( 1 - 2.69e3T + 2.86e7T^{2} \)
37 \( 1 + 7.25e3iT - 6.93e7T^{2} \)
41 \( 1 - 4.91e3T + 1.15e8T^{2} \)
43 \( 1 + 2.53e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.13e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.94e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.68e3T + 7.14e8T^{2} \)
61 \( 1 + 4.80e4T + 8.44e8T^{2} \)
67 \( 1 - 3.95e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.26e4T + 1.80e9T^{2} \)
73 \( 1 + 5.79e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.95e4T + 3.07e9T^{2} \)
83 \( 1 + 1.12e5iT - 3.93e9T^{2} \)
89 \( 1 + 6.69e4T + 5.58e9T^{2} \)
97 \( 1 - 1.31e5iT - 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.271071989354376194000172231828, −9.045511642220943264774909847057, −7.84860538773762113400349168160, −6.82036860451741879689070126072, −5.66762892036349972219549812629, −4.90553287078744234283765299790, −4.18973034651744355138241214951, −2.94128563501460717487056915411, −2.08428722872341190305178525225, −0.20014804276314120799015974411, 0.884653843208188171427550138838, 1.75483849318306169772232031473, 2.97731168492113993591196069098, 4.04266959197035143768955934205, 5.38133913275946709313441337319, 6.13392929513575756100835880359, 7.25806906788893424263516472831, 7.72846779517306396253657697428, 8.309336229235252548850781947218, 9.726138146052454470958259350030

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