Properties

Label 2-800-1.1-c3-0-4
Degree $2$
Conductor $800$
Sign $1$
Analytic cond. $47.2015$
Root an. cond. $6.87033$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.221·3-s − 22.7·7-s − 26.9·9-s − 66.8·11-s − 32.9·13-s + 35.9·17-s + 44.0·19-s + 5.04·21-s − 139.·23-s + 11.9·27-s + 134.·29-s + 229.·31-s + 14.8·33-s + 79.1·37-s + 7.29·39-s + 384.·41-s + 251.·43-s + 241.·47-s + 176.·49-s − 7.96·51-s − 222·53-s − 9.75·57-s − 552.·59-s + 494.·61-s + 614.·63-s − 574.·67-s + 30.7·69-s + ⋯
L(s)  = 1  − 0.0426·3-s − 1.23·7-s − 0.998·9-s − 1.83·11-s − 0.702·13-s + 0.512·17-s + 0.531·19-s + 0.0524·21-s − 1.26·23-s + 0.0851·27-s + 0.863·29-s + 1.32·31-s + 0.0780·33-s + 0.351·37-s + 0.0299·39-s + 1.46·41-s + 0.890·43-s + 0.749·47-s + 0.515·49-s − 0.0218·51-s − 0.575·53-s − 0.0226·57-s − 1.21·59-s + 1.03·61-s + 1.22·63-s − 1.04·67-s + 0.0537·69-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(47.2015\)
Root analytic conductor: \(6.87033\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7651809647\)
\(L(\frac12)\) \(\approx\) \(0.7651809647\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.221T + 27T^{2} \)
7 \( 1 + 22.7T + 343T^{2} \)
11 \( 1 + 66.8T + 1.33e3T^{2} \)
13 \( 1 + 32.9T + 2.19e3T^{2} \)
17 \( 1 - 35.9T + 4.91e3T^{2} \)
19 \( 1 - 44.0T + 6.85e3T^{2} \)
23 \( 1 + 139.T + 1.21e4T^{2} \)
29 \( 1 - 134.T + 2.43e4T^{2} \)
31 \( 1 - 229.T + 2.97e4T^{2} \)
37 \( 1 - 79.1T + 5.06e4T^{2} \)
41 \( 1 - 384.T + 6.89e4T^{2} \)
43 \( 1 - 251.T + 7.95e4T^{2} \)
47 \( 1 - 241.T + 1.03e5T^{2} \)
53 \( 1 + 222T + 1.48e5T^{2} \)
59 \( 1 + 552.T + 2.05e5T^{2} \)
61 \( 1 - 494.T + 2.26e5T^{2} \)
67 \( 1 + 574.T + 3.00e5T^{2} \)
71 \( 1 + 654.T + 3.57e5T^{2} \)
73 \( 1 + 1.13e3T + 3.89e5T^{2} \)
79 \( 1 - 179.T + 4.93e5T^{2} \)
83 \( 1 + 810.T + 5.71e5T^{2} \)
89 \( 1 + 29.7T + 7.04e5T^{2} \)
97 \( 1 - 383.T + 9.12e5T^{2} \)
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

−10.05720282814195846692261440805, −9.107916446581050191184195949292, −8.035447928550451228545283497791, −7.46673274893084027216199482250, −6.15443029925366657046983572283, −5.63504044744678489605424820924, −4.51064406338650598361290208100, −3.00732619171860983977566540048, −2.60811046616490515144911003721, −0.45592768968700238806794828979, 0.45592768968700238806794828979, 2.60811046616490515144911003721, 3.00732619171860983977566540048, 4.51064406338650598361290208100, 5.63504044744678489605424820924, 6.15443029925366657046983572283, 7.46673274893084027216199482250, 8.035447928550451228545283497791, 9.107916446581050191184195949292, 10.05720282814195846692261440805

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