Properties

Label 2-3800-1.1-c1-0-84
Degree $2$
Conductor $3800$
Sign $1$
Analytic cond. $30.3431$
Root an. cond. $5.50846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·7-s + 9-s − 4·11-s − 6·17-s − 19-s + 8·21-s − 8·23-s + 4·27-s − 6·29-s − 8·31-s + 8·33-s + 8·37-s − 2·41-s − 12·47-s + 9·49-s + 12·51-s − 4·53-s + 2·57-s + 8·59-s − 14·61-s − 4·63-s + 2·67-s + 16·69-s − 8·71-s + 2·73-s + 16·77-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.45·17-s − 0.229·19-s + 1.74·21-s − 1.66·23-s + 0.769·27-s − 1.11·29-s − 1.43·31-s + 1.39·33-s + 1.31·37-s − 0.312·41-s − 1.75·47-s + 9/7·49-s + 1.68·51-s − 0.549·53-s + 0.264·57-s + 1.04·59-s − 1.79·61-s − 0.503·63-s + 0.244·67-s + 1.92·69-s − 0.949·71-s + 0.234·73-s + 1.82·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(30.3431\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 3800,\ (\ :1/2),\ 1)\)

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 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−7.62470289893142858386832135132, −6.80908432518756936618354991115, −6.14592282549616514227497069696, −5.74407064504463232184891340466, −4.84120948838482838961825282274, −3.95618695370658675261359121213, −2.97883240342003233168458959111, −2.01847348961705052302389167550, 0, 0, 2.01847348961705052302389167550, 2.97883240342003233168458959111, 3.95618695370658675261359121213, 4.84120948838482838961825282274, 5.74407064504463232184891340466, 6.14592282549616514227497069696, 6.80908432518756936618354991115, 7.62470289893142858386832135132

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