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Properties

Label 2-800-1.1-c1-0-16
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 $1$

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: 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 \( 1 + p T^{2} \)
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

−9.911668703232740574424658064818, −8.772598804940102831000750865243, −8.422643539109821102866019859152, −7.09375449823698285987770578672, −6.47631929163503178303876770385, −5.26604485491629488374008831599, −4.52063183990333424217701740419, −3.11970456220302029413015085689, −2.12852635758913165860364844653, 0, 2.12852635758913165860364844653, 3.11970456220302029413015085689, 4.52063183990333424217701740419, 5.26604485491629488374008831599, 6.47631929163503178303876770385, 7.09375449823698285987770578672, 8.422643539109821102866019859152, 8.772598804940102831000750865243, 9.911668703232740574424658064818

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