Properties

Label 2-3920-1.1-c1-0-11
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 2·9-s + 6·11-s − 4·13-s + 15-s − 2·19-s + 3·23-s + 25-s + 5·27-s − 3·29-s − 8·31-s − 6·33-s − 4·37-s + 4·39-s + 9·41-s + 7·43-s + 2·45-s − 6·53-s − 6·55-s + 2·57-s + 6·59-s + 5·61-s + 4·65-s − 5·67-s − 3·69-s + 6·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s + 1.80·11-s − 1.10·13-s + 0.258·15-s − 0.458·19-s + 0.625·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 1.43·31-s − 1.04·33-s − 0.657·37-s + 0.640·39-s + 1.40·41-s + 1.06·43-s + 0.298·45-s − 0.824·53-s − 0.809·55-s + 0.264·57-s + 0.781·59-s + 0.640·61-s + 0.496·65-s − 0.610·67-s − 0.361·69-s + 0.712·71-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: yes
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.106254507\)
\(L(\frac12)\) \(\approx\) \(1.106254507\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.697733740758382740553611689848, −7.50284072082334084258384096942, −7.06135577446493813160311680830, −6.21080322340578476338876304348, −5.57381930747296822266094050342, −4.65687374438402552678393155173, −3.95953058177454500668475210908, −3.05422267129199327077040238038, −1.89700733491127948127375430876, −0.61766121148948183247449959512, 0.61766121148948183247449959512, 1.89700733491127948127375430876, 3.05422267129199327077040238038, 3.95953058177454500668475210908, 4.65687374438402552678393155173, 5.57381930747296822266094050342, 6.21080322340578476338876304348, 7.06135577446493813160311680830, 7.50284072082334084258384096942, 8.697733740758382740553611689848

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