Properties

Label 2-6480-1.1-c1-0-39
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·7-s + 3·11-s + 2·17-s − 19-s − 2·23-s + 25-s − 3·29-s + 3·31-s + 2·35-s + 5·41-s + 4·43-s + 8·47-s − 3·49-s + 2·53-s + 3·55-s − 3·59-s + 6·61-s + 10·67-s + 15·71-s − 14·73-s + 6·77-s + 8·79-s + 2·85-s + 89-s − 95-s − 16·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 0.904·11-s + 0.485·17-s − 0.229·19-s − 0.417·23-s + 1/5·25-s − 0.557·29-s + 0.538·31-s + 0.338·35-s + 0.780·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.404·55-s − 0.390·59-s + 0.768·61-s + 1.22·67-s + 1.78·71-s − 1.63·73-s + 0.683·77-s + 0.900·79-s + 0.216·85-s + 0.105·89-s − 0.102·95-s − 1.62·97-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

\(L(1)\) \(\approx\) \(2.735152851\)
\(L(\frac12)\) \(\approx\) \(2.735152851\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−8.063935604426200674224396391889, −7.31889550864010675363904262307, −6.58490919211320444242571076758, −5.86321794082847699930648840902, −5.24111283507903683306557369099, −4.34570066436012550998241994300, −3.74964088890026628300888660028, −2.62708931807601493968371165414, −1.79104930217700629611956524078, −0.905859705519913020792802597602, 0.905859705519913020792802597602, 1.79104930217700629611956524078, 2.62708931807601493968371165414, 3.74964088890026628300888660028, 4.34570066436012550998241994300, 5.24111283507903683306557369099, 5.86321794082847699930648840902, 6.58490919211320444242571076758, 7.31889550864010675363904262307, 8.063935604426200674224396391889

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