Properties

Label 2-6480-1.1-c1-0-34
Degree $2$
Conductor $6480$
Sign $1$
Analytic cond. $51.7430$
Root an. cond. $7.19326$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 3·7-s + 2·11-s − 2·13-s + 4·17-s + 8·19-s − 3·23-s + 25-s − 29-s − 3·35-s − 4·37-s + 5·41-s + 8·43-s − 7·47-s + 2·49-s − 2·53-s − 2·55-s + 14·59-s + 7·61-s + 2·65-s + 3·67-s − 2·71-s + 4·73-s + 6·77-s + 6·79-s − 9·83-s − 4·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.13·7-s + 0.603·11-s − 0.554·13-s + 0.970·17-s + 1.83·19-s − 0.625·23-s + 1/5·25-s − 0.185·29-s − 0.507·35-s − 0.657·37-s + 0.780·41-s + 1.21·43-s − 1.02·47-s + 2/7·49-s − 0.274·53-s − 0.269·55-s + 1.82·59-s + 0.896·61-s + 0.248·65-s + 0.366·67-s − 0.237·71-s + 0.468·73-s + 0.683·77-s + 0.675·79-s − 0.987·83-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6480\)    =    \(2^{4} \cdot 3^{4} \cdot 5\)
Sign: $1$
Analytic conductor: \(51.7430\)
Root analytic conductor: \(7.19326\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.393778280\)
\(L(\frac12)\) \(\approx\) \(2.393778280\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−7.990921188090353684329255048335, −7.40207927874331357562089818268, −6.82133651842454103936172110053, −5.61069069850025403436598361067, −5.28490659558466783587734238259, −4.37216752275413301188092512055, −3.68672752112726026926723145459, −2.80122069471657119570899805064, −1.69089801741608264928632165843, −0.857470382493630375229737718449, 0.857470382493630375229737718449, 1.69089801741608264928632165843, 2.80122069471657119570899805064, 3.68672752112726026926723145459, 4.37216752275413301188092512055, 5.28490659558466783587734238259, 5.61069069850025403436598361067, 6.82133651842454103936172110053, 7.40207927874331357562089818268, 7.990921188090353684329255048335

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