Properties

Label 2-3800-1.1-c1-0-45
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.53·3-s − 2.22·7-s − 0.652·9-s − 3.22·11-s + 1.57·13-s + 3.53·17-s + 19-s + 3.41·21-s + 4.47·23-s + 5.59·27-s + 1.92·29-s − 3.81·31-s + 4.94·33-s + 11.3·37-s − 2.41·39-s + 3.47·41-s − 1.69·43-s + 1.57·47-s − 2.04·49-s − 5.41·51-s − 7.12·53-s − 1.53·57-s − 7.88·59-s − 2.79·61-s + 1.45·63-s + 3.22·67-s − 6.85·69-s + ⋯
L(s)  = 1  − 0.884·3-s − 0.841·7-s − 0.217·9-s − 0.972·11-s + 0.436·13-s + 0.856·17-s + 0.229·19-s + 0.744·21-s + 0.933·23-s + 1.07·27-s + 0.356·29-s − 0.685·31-s + 0.860·33-s + 1.86·37-s − 0.386·39-s + 0.542·41-s − 0.258·43-s + 0.229·47-s − 0.291·49-s − 0.757·51-s − 0.979·53-s − 0.202·57-s − 1.02·59-s − 0.357·61-s + 0.183·63-s + 0.394·67-s − 0.825·69-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: \(1\)
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 \( 1 + 1.53T + 3T^{2} \)
7 \( 1 + 2.22T + 7T^{2} \)
11 \( 1 + 3.22T + 11T^{2} \)
13 \( 1 - 1.57T + 13T^{2} \)
17 \( 1 - 3.53T + 17T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 - 1.92T + 29T^{2} \)
31 \( 1 + 3.81T + 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 - 3.47T + 41T^{2} \)
43 \( 1 + 1.69T + 43T^{2} \)
47 \( 1 - 1.57T + 47T^{2} \)
53 \( 1 + 7.12T + 53T^{2} \)
59 \( 1 + 7.88T + 59T^{2} \)
61 \( 1 + 2.79T + 61T^{2} \)
67 \( 1 - 3.22T + 67T^{2} \)
71 \( 1 + 4.38T + 71T^{2} \)
73 \( 1 + 6.41T + 73T^{2} \)
79 \( 1 + 8.59T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 6.10T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−8.015256816270206099455471833494, −7.38200234923026199192949148490, −6.44088983363705150530449255541, −5.90712600264842395682485759048, −5.28366232558065771270557306381, −4.47236605857533314927640189871, −3.26686989886481837304472025698, −2.72581053548305303629416717507, −1.13527495865770761705306152221, 0, 1.13527495865770761705306152221, 2.72581053548305303629416717507, 3.26686989886481837304472025698, 4.47236605857533314927640189871, 5.28366232558065771270557306381, 5.90712600264842395682485759048, 6.44088983363705150530449255541, 7.38200234923026199192949148490, 8.015256816270206099455471833494

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