Properties

Label 2-800-1.1-c1-0-17
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.23·3-s − 5.23·7-s − 1.47·9-s − 6.47·21-s − 7.70·23-s − 5.52·27-s − 6·29-s + 4.47·41-s − 6.76·43-s − 0.291·47-s + 20.4·49-s + 13.4·61-s + 7.70·63-s − 14.1·67-s − 9.52·69-s − 2.41·81-s + 4.29·83-s − 7.41·87-s + 6·89-s − 18·101-s + 2.18·103-s + 19.7·107-s − 13.4·109-s + ⋯
L(s)  = 1  + 0.713·3-s − 1.97·7-s − 0.490·9-s − 1.41·21-s − 1.60·23-s − 1.06·27-s − 1.11·29-s + 0.698·41-s − 1.03·43-s − 0.0425·47-s + 2.91·49-s + 1.71·61-s + 0.971·63-s − 1.73·67-s − 1.14·69-s − 0.268·81-s + 0.471·83-s − 0.795·87-s + 0.635·89-s − 1.79·101-s + 0.214·103-s + 1.90·107-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: \(1\)
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 \( 1 - 1.23T + 3T^{2} \)
7 \( 1 + 5.23T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 7.70T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 6.76T + 43T^{2} \)
47 \( 1 + 0.291T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4.29T + 83T^{2} \)
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

−9.688425738060002106776785601423, −9.141449171057397419979971762855, −8.230332508152786921879558497199, −7.27148769963256059209523134586, −6.30124974466943294169167336688, −5.64347164095576435963526207442, −3.95966869533694550028977738025, −3.27365895702631250530520508134, −2.29274971104459976128018931335, 0, 2.29274971104459976128018931335, 3.27365895702631250530520508134, 3.95966869533694550028977738025, 5.64347164095576435963526207442, 6.30124974466943294169167336688, 7.27148769963256059209523134586, 8.230332508152786921879558497199, 9.141449171057397419979971762855, 9.688425738060002106776785601423

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