Properties

Label 2-3800-1.1-c1-0-53
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  + 2.68·3-s + 3.18·7-s + 4.18·9-s + 0.681·13-s + 1.18·17-s + 19-s + 8.55·21-s + 2.17·23-s + 3.18·27-s + 2.81·29-s + 6.37·31-s − 7.87·37-s + 1.82·39-s + 0.983·41-s − 1.36·43-s − 11.7·47-s + 3.17·49-s + 3.18·51-s + 1.69·53-s + 2.68·57-s + 11.5·59-s + 7.36·61-s + 13.3·63-s − 7.02·67-s + 5.82·69-s − 12.7·71-s + 5.53·73-s + ⋯
L(s)  = 1  + 1.54·3-s + 1.20·7-s + 1.39·9-s + 0.188·13-s + 0.288·17-s + 0.229·19-s + 1.86·21-s + 0.453·23-s + 0.613·27-s + 0.521·29-s + 1.14·31-s − 1.29·37-s + 0.292·39-s + 0.153·41-s − 0.207·43-s − 1.71·47-s + 0.453·49-s + 0.446·51-s + 0.233·53-s + 0.355·57-s + 1.50·59-s + 0.942·61-s + 1.68·63-s − 0.858·67-s + 0.701·69-s − 1.51·71-s + 0.647·73-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

\(L(1)\) \(\approx\) \(4.157462200\)
\(L(\frac12)\) \(\approx\) \(4.157462200\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.68T + 3T^{2} \)
7 \( 1 - 3.18T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 0.681T + 13T^{2} \)
17 \( 1 - 1.18T + 17T^{2} \)
23 \( 1 - 2.17T + 23T^{2} \)
29 \( 1 - 2.81T + 29T^{2} \)
31 \( 1 - 6.37T + 31T^{2} \)
37 \( 1 + 7.87T + 37T^{2} \)
41 \( 1 - 0.983T + 41T^{2} \)
43 \( 1 + 1.36T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 1.69T + 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 - 7.36T + 61T^{2} \)
67 \( 1 + 7.02T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 - 5.53T + 73T^{2} \)
79 \( 1 - 5.36T + 79T^{2} \)
83 \( 1 + 2.37T + 83T^{2} \)
89 \( 1 - 3.01T + 89T^{2} \)
97 \( 1 + 4.88T + 97T^{2} \)
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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

−8.511450908488090642244650931466, −7.945885488587093715408326221859, −7.32473634266634908536826590450, −6.47112873718190363658502906771, −5.27366477963395829838655696950, −4.62540801797066833083248161955, −3.71062852587024621764288342608, −2.96081568062518995074100532287, −2.07285257420737636423374911168, −1.23386602588461115297009009683, 1.23386602588461115297009009683, 2.07285257420737636423374911168, 2.96081568062518995074100532287, 3.71062852587024621764288342608, 4.62540801797066833083248161955, 5.27366477963395829838655696950, 6.47112873718190363658502906771, 7.32473634266634908536826590450, 7.945885488587093715408326221859, 8.511450908488090642244650931466

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