↑

Properties

Label 2-800-5.4-c3-0-39
Degree $2$
Conductor $800$
Sign $0.447 + 0.894i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.221i·3-s − 22.7i·7-s + 26.9·9-s + 66.8·11-s − 32.9i·13-s − 35.9i·17-s + 44.0·19-s + 5.04·21-s + 139. i·23-s + 11.9i·27-s − 134.·29-s − 229.·31-s + 14.8i·33-s − 79.1i·37-s + 7.29·39-s + ⋯
L(s)  = 1  + 0.0426i·3-s − 1.23i·7-s + 0.998·9-s + 1.83·11-s − 0.702i·13-s − 0.512i·17-s + 0.531·19-s + 0.0524·21-s + 1.26i·23-s + 0.0851i·27-s − 0.863·29-s − 1.32·31-s + 0.0780i·33-s − 0.351i·37-s + 0.0299·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(47.2015\)
Root analytic conductor: \(6.87033\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.477158007\)
\(L(\frac12)\) \(\approx\) \(2.477158007\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.221iT - 27T^{2} \)
7 \( 1 + 22.7iT - 343T^{2} \)
11 \( 1 - 66.8T + 1.33e3T^{2} \)
13 \( 1 + 32.9iT - 2.19e3T^{2} \)
17 \( 1 + 35.9iT - 4.91e3T^{2} \)
19 \( 1 - 44.0T + 6.85e3T^{2} \)
23 \( 1 - 139. iT - 1.21e4T^{2} \)
29 \( 1 + 134.T + 2.43e4T^{2} \)
31 \( 1 + 229.T + 2.97e4T^{2} \)
37 \( 1 + 79.1iT - 5.06e4T^{2} \)
41 \( 1 - 384.T + 6.89e4T^{2} \)
43 \( 1 + 251. iT - 7.95e4T^{2} \)
47 \( 1 - 241. iT - 1.03e5T^{2} \)
53 \( 1 + 222iT - 1.48e5T^{2} \)
59 \( 1 + 552.T + 2.05e5T^{2} \)
61 \( 1 - 494.T + 2.26e5T^{2} \)
67 \( 1 + 574. iT - 3.00e5T^{2} \)
71 \( 1 - 654.T + 3.57e5T^{2} \)
73 \( 1 + 1.13e3iT - 3.89e5T^{2} \)
79 \( 1 - 179.T + 4.93e5T^{2} \)
83 \( 1 - 810. iT - 5.71e5T^{2} \)
89 \( 1 - 29.7T + 7.04e5T^{2} \)
97 \( 1 + 383. iT - 9.12e5T^{2} \)
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

−9.493976875993539913074282384607, −9.295898124301121679560560543881, −7.64060627942133489626561206208, −7.31357227000724683078943733871, −6.37923920419345871208945520041, −5.19059502996193960623832001955, −3.98580960367699463716697391757, −3.59234836999917873547731093525, −1.65217422339950666728788142448, −0.76430745300423087283678873326, 1.26665333210474088595075721804, 2.21669384813536641245880321668, 3.71963960621793145932268036255, 4.48529433789378771525063074415, 5.75550365947288292884520778832, 6.53096562114867678543874132510, 7.30416320569295927617845213698, 8.557232012095973567324745222855, 9.210950259885002464144588746023, 9.746318277310583470940232942847

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