Properties

Label 2-3920-1.1-c1-0-45
Degree $2$
Conductor $3920$
Sign $-1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
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  − 3.41·3-s + 5-s + 8.65·9-s + 0.828·11-s − 4.82·13-s − 3.41·15-s + 2.58·17-s − 0.585·19-s + 1.17·23-s + 25-s − 19.3·27-s − 4.82·29-s + 2.82·31-s − 2.82·33-s − 7.65·37-s + 16.4·39-s − 3.07·41-s + 8.82·43-s + 8.65·45-s − 5.17·47-s − 8.82·51-s + 6.48·53-s + 0.828·55-s + 2·57-s − 8.58·59-s + 9.31·61-s − 4.82·65-s + ⋯
L(s)  = 1  − 1.97·3-s + 0.447·5-s + 2.88·9-s + 0.249·11-s − 1.33·13-s − 0.881·15-s + 0.627·17-s − 0.134·19-s + 0.244·23-s + 0.200·25-s − 3.71·27-s − 0.896·29-s + 0.508·31-s − 0.492·33-s − 1.25·37-s + 2.63·39-s − 0.479·41-s + 1.34·43-s + 1.29·45-s − 0.754·47-s − 1.23·51-s + 0.890·53-s + 0.111·55-s + 0.264·57-s − 1.11·59-s + 1.19·61-s − 0.598·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
good3 \( 1 + 3.41T + 3T^{2} \)
11 \( 1 - 0.828T + 11T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 - 2.58T + 17T^{2} \)
19 \( 1 + 0.585T + 19T^{2} \)
23 \( 1 - 1.17T + 23T^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 + 3.07T + 41T^{2} \)
43 \( 1 - 8.82T + 43T^{2} \)
47 \( 1 + 5.17T + 47T^{2} \)
53 \( 1 - 6.48T + 53T^{2} \)
59 \( 1 + 8.58T + 59T^{2} \)
61 \( 1 - 9.31T + 61T^{2} \)
67 \( 1 + 1.65T + 67T^{2} \)
71 \( 1 - 4.48T + 71T^{2} \)
73 \( 1 + 9.41T + 73T^{2} \)
79 \( 1 - 6.82T + 79T^{2} \)
83 \( 1 - 2.24T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 7.75T + 97T^{2} \)
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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

−7.75725520800288412737175554622, −7.15251506997773743161281959086, −6.53516021015352282878328312876, −5.78767811224341639216595916388, −5.19131395441803621721540178762, −4.66307881814671215048982783548, −3.67165968245932620244619209745, −2.19287258485472466293923471795, −1.15834579324384561015485594743, 0, 1.15834579324384561015485594743, 2.19287258485472466293923471795, 3.67165968245932620244619209745, 4.66307881814671215048982783548, 5.19131395441803621721540178762, 5.78767811224341639216595916388, 6.53516021015352282878328312876, 7.15251506997773743161281959086, 7.75725520800288412737175554622

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