Properties

Label 2-3800-1.1-c1-0-35
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  − 3.29·3-s − 1.78·7-s + 7.87·9-s − 5.08·11-s + 1.29·13-s − 0.213·17-s − 19-s + 5.89·21-s + 3.72·23-s − 16.0·27-s + 0.870·29-s + 16.7·33-s + 2·37-s − 4.27·39-s + 8.59·41-s + 3.67·43-s + 4.65·47-s − 3.80·49-s + 0.702·51-s − 11.0·53-s + 3.29·57-s − 4.70·59-s + 3.51·61-s − 14.0·63-s − 1.12·67-s − 12.2·69-s + 8.76·71-s + ⋯
L(s)  = 1  − 1.90·3-s − 0.675·7-s + 2.62·9-s − 1.53·11-s + 0.359·13-s − 0.0517·17-s − 0.229·19-s + 1.28·21-s + 0.776·23-s − 3.09·27-s + 0.161·29-s + 2.91·33-s + 0.328·37-s − 0.684·39-s + 1.34·41-s + 0.560·43-s + 0.679·47-s − 0.543·49-s + 0.0984·51-s − 1.51·53-s + 0.436·57-s − 0.612·59-s + 0.449·61-s − 1.77·63-s − 0.137·67-s − 1.47·69-s + 1.03·71-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 \( 1 + 3.29T + 3T^{2} \)
7 \( 1 + 1.78T + 7T^{2} \)
11 \( 1 + 5.08T + 11T^{2} \)
13 \( 1 - 1.29T + 13T^{2} \)
17 \( 1 + 0.213T + 17T^{2} \)
23 \( 1 - 3.72T + 23T^{2} \)
29 \( 1 - 0.870T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 8.59T + 41T^{2} \)
43 \( 1 - 3.67T + 43T^{2} \)
47 \( 1 - 4.65T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 + 4.70T + 59T^{2} \)
61 \( 1 - 3.51T + 61T^{2} \)
67 \( 1 + 1.12T + 67T^{2} \)
71 \( 1 - 8.76T + 71T^{2} \)
73 \( 1 + 6.80T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 - 9.74T + 83T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 + 4.16T + 97T^{2} \)
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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

−7.80381217975699637900943138792, −7.30967895461495821502838060420, −6.31464764747388378325647371128, −6.04781196742289302687109806718, −5.11938228523696879514672314178, −4.68816576846199661474043952688, −3.59496429574839096142811080470, −2.40883194911886275358117557305, −0.991902903306918220094810679184, 0, 0.991902903306918220094810679184, 2.40883194911886275358117557305, 3.59496429574839096142811080470, 4.68816576846199661474043952688, 5.11938228523696879514672314178, 6.04781196742289302687109806718, 6.31464764747388378325647371128, 7.30967895461495821502838060420, 7.80381217975699637900943138792

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