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Properties

Label 2-6480-1.1-c1-0-67
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 + 7-s − 4·13-s + 6·17-s − 2·19-s − 3·23-s + 25-s − 3·29-s + 10·31-s − 35-s − 10·37-s − 9·41-s + 4·43-s + 9·47-s − 6·49-s + 6·53-s − 6·59-s − 61-s + 4·65-s − 11·67-s + 12·71-s − 4·73-s + 10·79-s − 9·83-s − 6·85-s − 9·89-s − 4·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 1.10·13-s + 1.45·17-s − 0.458·19-s − 0.625·23-s + 1/5·25-s − 0.557·29-s + 1.79·31-s − 0.169·35-s − 1.64·37-s − 1.40·41-s + 0.609·43-s + 1.31·47-s − 6/7·49-s + 0.824·53-s − 0.781·59-s − 0.128·61-s + 0.496·65-s − 1.34·67-s + 1.42·71-s − 0.468·73-s + 1.12·79-s − 0.987·83-s − 0.650·85-s − 0.953·89-s − 0.419·91-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 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.67252102220871873654798706834, −7.11940944406184087615315155008, −6.27143814897454695539992376547, −5.41022613855701299790356949391, −4.83473355489696479578744856453, −4.02282497998374753778687713943, −3.21358016736820918032737686953, −2.33193074993550367101986670871, −1.28941564782313038107017061315, 0, 1.28941564782313038107017061315, 2.33193074993550367101986670871, 3.21358016736820918032737686953, 4.02282497998374753778687713943, 4.83473355489696479578744856453, 5.41022613855701299790356949391, 6.27143814897454695539992376547, 7.11940944406184087615315155008, 7.67252102220871873654798706834

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