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Properties

Label 2-3920-20.19-c0-0-5
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

\(L(\frac{1}{2})\) \(\approx\) \(2.107326376\)
\(L(\frac12)\) \(\approx\) \(2.107326376\)
\(L(1)\) 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

−8.670518571064718181213019787662, −7.948908863248564490121492483308, −7.35407386036474982897199637872, −6.92373550296238589167841172083, −5.59242626274693491723804348987, −4.47862804396335827529305299116, −3.94997644358720342591134005476, −3.10620335820860592271843950989, −2.54316683244857705306886928076, −1.25704503398957430265168342870, 1.25704503398957430265168342870, 2.54316683244857705306886928076, 3.10620335820860592271843950989, 3.94997644358720342591134005476, 4.47862804396335827529305299116, 5.59242626274693491723804348987, 6.92373550296238589167841172083, 7.35407386036474982897199637872, 7.948908863248564490121492483308, 8.670518571064718181213019787662

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