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Properties

Label 2-3800-1.1-c1-0-77
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s − 4·11-s − 4·13-s + 2·17-s + 19-s + 4·23-s − 4·27-s − 6·29-s − 8·31-s − 8·33-s + 4·37-s − 8·39-s − 2·41-s − 4·43-s + 8·47-s − 7·49-s + 4·51-s + 2·57-s − 8·59-s + 2·61-s + 14·67-s + 8·69-s − 8·71-s − 6·73-s − 4·79-s − 11·81-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.485·17-s + 0.229·19-s + 0.834·23-s − 0.769·27-s − 1.11·29-s − 1.43·31-s − 1.39·33-s + 0.657·37-s − 1.28·39-s − 0.312·41-s − 0.609·43-s + 1.16·47-s − 49-s + 0.560·51-s + 0.264·57-s − 1.04·59-s + 0.256·61-s + 1.71·67-s + 0.963·69-s − 0.949·71-s − 0.702·73-s − 0.450·79-s − 1.22·81-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: \(1\)
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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.057957786137874692259859316773, −7.53910007236387439081696157757, −7.02342470351221686278123872915, −5.65823260358416954344476041772, −5.20755635438023146479382322783, −4.16014067026549737843355106345, −3.17458963214210180570356775634, −2.66236472835453567561224781182, −1.74567787118854747318287118710, 0, 1.74567787118854747318287118710, 2.66236472835453567561224781182, 3.17458963214210180570356775634, 4.16014067026549737843355106345, 5.20755635438023146479382322783, 5.65823260358416954344476041772, 7.02342470351221686278123872915, 7.53910007236387439081696157757, 8.057957786137874692259859316773

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