Properties

Label 2-6480-1.1-c1-0-65
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s − 3·11-s − 4·13-s + 6·17-s + 19-s − 6·23-s + 25-s + 9·29-s + 31-s − 2·35-s + 8·37-s − 3·41-s + 4·43-s + 12·47-s − 3·49-s − 6·53-s − 3·55-s + 3·59-s − 10·61-s − 4·65-s − 14·67-s − 3·71-s + 2·73-s + 6·77-s + 16·79-s − 12·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.904·11-s − 1.10·13-s + 1.45·17-s + 0.229·19-s − 1.25·23-s + 1/5·25-s + 1.67·29-s + 0.179·31-s − 0.338·35-s + 1.31·37-s − 0.468·41-s + 0.609·43-s + 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.404·55-s + 0.390·59-s − 1.28·61-s − 0.496·65-s − 1.71·67-s − 0.356·71-s + 0.234·73-s + 0.683·77-s + 1.80·79-s − 1.31·83-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: \(1\)
Selberg data: \((2,\ 6480,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.77998618695953720594535168840, −6.97470864030422216883135766254, −6.14327731863010235437310253551, −5.62125020456127945130841137656, −4.86736620926560928394752901062, −4.03682599286201905184942890207, −2.86723578394976962668175731197, −2.62322801373417856633469243239, −1.26374294638807307583380893960, 0, 1.26374294638807307583380893960, 2.62322801373417856633469243239, 2.86723578394976962668175731197, 4.03682599286201905184942890207, 4.86736620926560928394752901062, 5.62125020456127945130841137656, 6.14327731863010235437310253551, 6.97470864030422216883135766254, 7.77998618695953720594535168840

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