Properties

Label 2-800-1.1-c1-0-4
Degree $2$
Conductor $800$
Sign $1$
Analytic cond. $6.38803$
Root an. cond. $2.52745$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 4·13-s + 8·17-s + 10·29-s + 12·37-s − 10·41-s − 7·49-s − 4·53-s + 10·61-s + 16·73-s + 9·81-s − 10·89-s − 8·97-s − 2·101-s + 6·109-s − 16·113-s − 12·117-s + ⋯
L(s)  = 1  − 9-s + 1.10·13-s + 1.94·17-s + 1.85·29-s + 1.97·37-s − 1.56·41-s − 49-s − 0.549·53-s + 1.28·61-s + 1.87·73-s + 81-s − 1.05·89-s − 0.812·97-s − 0.199·101-s + 0.574·109-s − 1.50·113-s − 1.10·117-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: yes
Arithmetic: yes
Character: $\chi_{800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.568356227\)
\(L(\frac12)\) \(\approx\) \(1.568356227\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−10.23438680846563734808416701606, −9.481001567833775866135833367998, −8.318458181732681568361082997175, −8.043199677138800978749971209938, −6.64037527171448229645548827286, −5.88600838036285002216361121358, −5.01663774178771706139531826817, −3.66097610339341905494433801925, −2.81202536212746604283359988203, −1.09831956032328018823451631467, 1.09831956032328018823451631467, 2.81202536212746604283359988203, 3.66097610339341905494433801925, 5.01663774178771706139531826817, 5.88600838036285002216361121358, 6.64037527171448229645548827286, 8.043199677138800978749971209938, 8.318458181732681568361082997175, 9.481001567833775866135833367998, 10.23438680846563734808416701606

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