Properties

Label 2-800-1.1-c1-0-9
Degree $2$
Conductor $800$
Sign $-1$
Analytic cond. $6.38803$
Root an. cond. $2.52745$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.23·3-s − 0.763·7-s + 7.47·9-s + 2.47·21-s + 5.70·23-s − 14.4·27-s − 6·29-s − 4.47·41-s − 11.2·43-s − 13.7·47-s − 6.41·49-s − 13.4·61-s − 5.70·63-s + 8.18·67-s − 18.4·69-s + 24.4·81-s + 17.7·83-s + 19.4·87-s + 6·89-s − 18·101-s − 20.1·103-s + 6.29·107-s + 13.4·109-s + ⋯
L(s)  = 1  − 1.86·3-s − 0.288·7-s + 2.49·9-s + 0.539·21-s + 1.19·23-s − 2.78·27-s − 1.11·29-s − 0.698·41-s − 1.71·43-s − 1.99·47-s − 0.916·49-s − 1.71·61-s − 0.719·63-s + 0.999·67-s − 2.22·69-s + 2.71·81-s + 1.94·83-s + 2.08·87-s + 0.635·89-s − 1.79·101-s − 1.98·103-s + 0.608·107-s + 1.28·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.23T + 3T^{2} \)
7 \( 1 + 0.763T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 5.70T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + 13.7T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 - 8.18T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 17.7T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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

−10.03737833035502482587801863851, −9.300582179241337954509875615970, −7.924778022804316373882635712297, −6.85684343518019557369700046443, −6.38706411435082990241653706437, −5.32430888987114368555108609273, −4.77612870875324925388058545659, −3.47910078778398930143432965745, −1.52217958752794764656562061729, 0, 1.52217958752794764656562061729, 3.47910078778398930143432965745, 4.77612870875324925388058545659, 5.32430888987114368555108609273, 6.38706411435082990241653706437, 6.85684343518019557369700046443, 7.924778022804316373882635712297, 9.300582179241337954509875615970, 10.03737833035502482587801863851

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