Properties

Label 2-800-1.1-c1-0-12
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 $0$

Origins

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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: $\chi_{800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.981715976\)
\(L(\frac12)\) \(\approx\) \(2.981715976\)
\(L(\frac{3}{2})\) 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 \( 1 + 97T^{2} \)
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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

−10.00682755962414974756417550010, −9.249558785995916752160956436642, −8.621293987761356855561274600126, −7.75915321252701140688166020057, −7.26816707798868962597348483819, −5.92584847778616959918884557427, −4.47686394719811505501406495279, −3.71031466964137012738153082018, −2.64624472074562449511516543052, −1.67669514643215980846346607477, 1.67669514643215980846346607477, 2.64624472074562449511516543052, 3.71031466964137012738153082018, 4.47686394719811505501406495279, 5.92584847778616959918884557427, 7.26816707798868962597348483819, 7.75915321252701140688166020057, 8.621293987761356855561274600126, 9.249558785995916752160956436642, 10.00682755962414974756417550010

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