Properties

Label 2-3920-20.19-c0-0-3
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·3-s + 5-s + 1.99·9-s − 1.73·15-s + 1.73·23-s + 25-s − 1.73·27-s − 29-s − 41-s + 1.73·43-s + 1.99·45-s − 61-s − 1.73·67-s − 2.99·69-s − 1.73·75-s + 0.999·81-s + 1.73·83-s + 1.73·87-s + 89-s + 101-s + 1.73·103-s − 1.73·107-s + 109-s + 1.73·115-s + ⋯
L(s)  = 1  − 1.73·3-s + 5-s + 1.99·9-s − 1.73·15-s + 1.73·23-s + 25-s − 1.73·27-s − 29-s − 41-s + 1.73·43-s + 1.99·45-s − 61-s − 1.73·67-s − 2.99·69-s − 1.73·75-s + 0.999·81-s + 1.73·83-s + 1.73·87-s + 89-s + 101-s + 1.73·103-s − 1.73·107-s + 109-s + 1.73·115-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (3039, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8958430096\)
\(L(\frac12)\) \(\approx\) \(0.8958430096\)
\(L(1)\) 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 \( 1 + 1.73T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.73T + T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + 1.73T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.73T + T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.956108210562206209990784044381, −7.59514839199470243006805898537, −6.96165348348636394809829758425, −6.26634060818155616380010169127, −5.70779353000963306177534561303, −5.07946398376705954394716485155, −4.46792955475968031660661767907, −3.18649694421326464983934446220, −1.91424914110462113324793165761, −0.915010044279353065802582012378, 0.915010044279353065802582012378, 1.91424914110462113324793165761, 3.18649694421326464983934446220, 4.46792955475968031660661767907, 5.07946398376705954394716485155, 5.70779353000963306177534561303, 6.26634060818155616380010169127, 6.96165348348636394809829758425, 7.59514839199470243006805898537, 8.956108210562206209990784044381

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