Properties

Degree $2$
Conductor $3920$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s + 9-s − 3·11-s + 5·13-s − 2·15-s + 6·17-s + 19-s − 3·23-s + 25-s − 4·27-s − 6·29-s + 4·31-s − 6·33-s + 11·37-s + 10·39-s + 3·41-s + 10·43-s − 45-s − 3·47-s + 12·51-s + 3·53-s + 3·55-s + 2·57-s − 4·61-s − 5·65-s + 4·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 1.38·13-s − 0.516·15-s + 1.45·17-s + 0.229·19-s − 0.625·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 1.04·33-s + 1.80·37-s + 1.60·39-s + 0.468·41-s + 1.52·43-s − 0.149·45-s − 0.437·47-s + 1.68·51-s + 0.412·53-s + 0.404·55-s + 0.264·57-s − 0.512·61-s − 0.620·65-s + 0.488·67-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: $1$
Motivic weight: \(1\)
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.727587534\)
\(L(\frac12)\) \(\approx\) \(2.727587534\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.312353373911425463378423277597, −7.81189036994320081381812287447, −7.47025394478870762896015822967, −6.09431538082891075849115819441, −5.65372048903059002115277189268, −4.45082680945347931104217752177, −3.63096321073868899269924714815, −3.08375678221313543410500442005, −2.17787868933866687584737323067, −0.919024052519317686191599815600, 0.919024052519317686191599815600, 2.17787868933866687584737323067, 3.08375678221313543410500442005, 3.63096321073868899269924714815, 4.45082680945347931104217752177, 5.65372048903059002115277189268, 6.09431538082891075849115819441, 7.47025394478870762896015822967, 7.81189036994320081381812287447, 8.312353373911425463378423277597

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