Properties

Label 2-3800-1.1-c1-0-9
Degree $2$
Conductor $3800$
Sign $1$
Analytic cond. $30.3431$
Root an. cond. $5.50846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.32·3-s + 1.39·7-s + 2.39·9-s − 4.32·13-s − 0.601·17-s + 19-s − 3.24·21-s − 6.04·23-s + 1.39·27-s + 4.60·29-s + 2.79·31-s − 1.07·37-s + 10.0·39-s − 5.44·41-s + 8.64·43-s + 1.85·47-s − 5.04·49-s + 1.39·51-s + 3.11·53-s − 2.32·57-s − 6.69·59-s − 2.64·61-s + 3.35·63-s + 14.4·67-s + 14.0·69-s − 5.59·71-s − 12.6·73-s + ⋯
L(s)  = 1  − 1.34·3-s + 0.528·7-s + 0.799·9-s − 1.19·13-s − 0.145·17-s + 0.229·19-s − 0.708·21-s − 1.26·23-s + 0.269·27-s + 0.854·29-s + 0.502·31-s − 0.176·37-s + 1.60·39-s − 0.850·41-s + 1.31·43-s + 0.269·47-s − 0.720·49-s + 0.195·51-s + 0.428·53-s − 0.307·57-s − 0.871·59-s − 0.338·61-s + 0.422·63-s + 1.76·67-s + 1.69·69-s − 0.663·71-s − 1.48·73-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8528094979\)
\(L(\frac12)\) \(\approx\) \(0.8528094979\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.32T + 3T^{2} \)
7 \( 1 - 1.39T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4.32T + 13T^{2} \)
17 \( 1 + 0.601T + 17T^{2} \)
23 \( 1 + 6.04T + 23T^{2} \)
29 \( 1 - 4.60T + 29T^{2} \)
31 \( 1 - 2.79T + 31T^{2} \)
37 \( 1 + 1.07T + 37T^{2} \)
41 \( 1 + 5.44T + 41T^{2} \)
43 \( 1 - 8.64T + 43T^{2} \)
47 \( 1 - 1.85T + 47T^{2} \)
53 \( 1 - 3.11T + 53T^{2} \)
59 \( 1 + 6.69T + 59T^{2} \)
61 \( 1 + 2.64T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 + 5.59T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 4.64T + 79T^{2} \)
83 \( 1 - 1.20T + 83T^{2} \)
89 \( 1 - 9.44T + 89T^{2} \)
97 \( 1 + 4.51T + 97T^{2} \)
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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

−8.380616138503332224760077852312, −7.66971334114824390960454798374, −6.91571282832885929961100313544, −6.18722576639986749341417432691, −5.50277231845999237152382609342, −4.80589408029310738882199073397, −4.24965189345675049307551633763, −2.90132907773152857585560505923, −1.83059318664171725861266230789, −0.56704580160764500942140021190, 0.56704580160764500942140021190, 1.83059318664171725861266230789, 2.90132907773152857585560505923, 4.24965189345675049307551633763, 4.80589408029310738882199073397, 5.50277231845999237152382609342, 6.18722576639986749341417432691, 6.91571282832885929961100313544, 7.66971334114824390960454798374, 8.380616138503332224760077852312

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