Properties

Label 2-3800-1.1-c1-0-63
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  − 0.664·3-s + 0.345·7-s − 2.55·9-s + 3.51·11-s + 3.22·13-s − 4.88·17-s + 19-s − 0.229·21-s − 7.62·23-s + 3.69·27-s − 1.73·29-s − 4.76·31-s − 2.33·33-s + 1.86·37-s − 2.14·39-s + 6.76·41-s − 0.606·43-s − 3.34·47-s − 6.88·49-s + 3.24·51-s + 0.107·53-s − 0.664·57-s + 12.3·59-s − 8.41·61-s − 0.884·63-s − 2.23·67-s + 5.06·69-s + ⋯
L(s)  = 1  − 0.383·3-s + 0.130·7-s − 0.852·9-s + 1.06·11-s + 0.894·13-s − 1.18·17-s + 0.229·19-s − 0.0501·21-s − 1.59·23-s + 0.710·27-s − 0.323·29-s − 0.855·31-s − 0.406·33-s + 0.307·37-s − 0.343·39-s + 1.05·41-s − 0.0924·43-s − 0.488·47-s − 0.982·49-s + 0.454·51-s + 0.0147·53-s − 0.0880·57-s + 1.60·59-s − 1.07·61-s − 0.111·63-s − 0.273·67-s + 0.610·69-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 \( 1 + 0.664T + 3T^{2} \)
7 \( 1 - 0.345T + 7T^{2} \)
11 \( 1 - 3.51T + 11T^{2} \)
13 \( 1 - 3.22T + 13T^{2} \)
17 \( 1 + 4.88T + 17T^{2} \)
23 \( 1 + 7.62T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 + 4.76T + 31T^{2} \)
37 \( 1 - 1.86T + 37T^{2} \)
41 \( 1 - 6.76T + 41T^{2} \)
43 \( 1 + 0.606T + 43T^{2} \)
47 \( 1 + 3.34T + 47T^{2} \)
53 \( 1 - 0.107T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 + 8.41T + 61T^{2} \)
67 \( 1 + 2.23T + 67T^{2} \)
71 \( 1 + 0.536T + 71T^{2} \)
73 \( 1 + 5.57T + 73T^{2} \)
79 \( 1 + 2.67T + 79T^{2} \)
83 \( 1 + 1.79T + 83T^{2} \)
89 \( 1 - 8.94T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−8.245098151402001525541520375792, −7.38258184101036106601574655264, −6.32538285869862024986811479092, −6.14835844746175876651844696368, −5.19472068512857797717119603548, −4.20519312046796208485193517784, −3.60768974297536016359389490388, −2.42444030761923594410837368434, −1.40249609117630693939587679186, 0, 1.40249609117630693939587679186, 2.42444030761923594410837368434, 3.60768974297536016359389490388, 4.20519312046796208485193517784, 5.19472068512857797717119603548, 6.14835844746175876651844696368, 6.32538285869862024986811479092, 7.38258184101036106601574655264, 8.245098151402001525541520375792

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