Properties

Label 2-800-1.1-c3-0-37
Degree $2$
Conductor $800$
Sign $-1$
Analytic cond. $47.2015$
Root an. cond. $6.87033$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s + 10·7-s − 2·9-s − 15·11-s + 8·13-s − 21·17-s + 105·19-s − 50·21-s + 10·23-s + 145·27-s − 20·29-s − 230·31-s + 75·33-s − 54·37-s − 40·39-s − 195·41-s + 300·43-s + 480·47-s − 243·49-s + 105·51-s + 322·53-s − 525·57-s + 560·59-s − 730·61-s − 20·63-s − 255·67-s − 50·69-s + ⋯
L(s)  = 1  − 0.962·3-s + 0.539·7-s − 0.0740·9-s − 0.411·11-s + 0.170·13-s − 0.299·17-s + 1.26·19-s − 0.519·21-s + 0.0906·23-s + 1.03·27-s − 0.128·29-s − 1.33·31-s + 0.395·33-s − 0.239·37-s − 0.164·39-s − 0.742·41-s + 1.06·43-s + 1.48·47-s − 0.708·49-s + 0.288·51-s + 0.834·53-s − 1.21·57-s + 1.23·59-s − 1.53·61-s − 0.0399·63-s − 0.464·67-s − 0.0872·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(47.2015\)
Root analytic conductor: \(6.87033\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 800,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 5 T + p^{3} T^{2} \)
7 \( 1 - 10 T + p^{3} T^{2} \)
11 \( 1 + 15 T + p^{3} T^{2} \)
13 \( 1 - 8 T + p^{3} T^{2} \)
17 \( 1 + 21 T + p^{3} T^{2} \)
19 \( 1 - 105 T + p^{3} T^{2} \)
23 \( 1 - 10 T + p^{3} T^{2} \)
29 \( 1 + 20 T + p^{3} T^{2} \)
31 \( 1 + 230 T + p^{3} T^{2} \)
37 \( 1 + 54 T + p^{3} T^{2} \)
41 \( 1 + 195 T + p^{3} T^{2} \)
43 \( 1 - 300 T + p^{3} T^{2} \)
47 \( 1 - 480 T + p^{3} T^{2} \)
53 \( 1 - 322 T + p^{3} T^{2} \)
59 \( 1 - 560 T + p^{3} T^{2} \)
61 \( 1 + 730 T + p^{3} T^{2} \)
67 \( 1 + 255 T + p^{3} T^{2} \)
71 \( 1 + 40 T + p^{3} T^{2} \)
73 \( 1 - 317 T + p^{3} T^{2} \)
79 \( 1 + 830 T + p^{3} T^{2} \)
83 \( 1 + 75 T + p^{3} T^{2} \)
89 \( 1 + 705 T + p^{3} T^{2} \)
97 \( 1 + 1434 T + p^{3} T^{2} \)
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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

−9.494265582471169569084386928675, −8.611024475443215609998692411611, −7.62837865766034788794997705799, −6.81693389753055143431240344542, −5.62425802921389752899918836430, −5.27760518793107670337513667707, −4.08652793072684134919719691256, −2.75903444728083271328081954383, −1.30109440170486931385299102791, 0, 1.30109440170486931385299102791, 2.75903444728083271328081954383, 4.08652793072684134919719691256, 5.27760518793107670337513667707, 5.62425802921389752899918836430, 6.81693389753055143431240344542, 7.62837865766034788794997705799, 8.611024475443215609998692411611, 9.494265582471169569084386928675

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