Properties

Label 2-6480-1.1-c1-0-82
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 − 5·11-s + 4·13-s − 4·17-s + 5·19-s − 6·23-s + 25-s − 5·29-s + 9·31-s − 10·37-s + 7·41-s + 2·43-s − 2·47-s − 7·49-s + 8·53-s − 5·55-s + 59-s − 2·61-s + 4·65-s − 6·67-s − 71-s − 8·73-s − 12·79-s − 6·83-s − 4·85-s − 9·89-s + 5·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.50·11-s + 1.10·13-s − 0.970·17-s + 1.14·19-s − 1.25·23-s + 1/5·25-s − 0.928·29-s + 1.61·31-s − 1.64·37-s + 1.09·41-s + 0.304·43-s − 0.291·47-s − 49-s + 1.09·53-s − 0.674·55-s + 0.130·59-s − 0.256·61-s + 0.496·65-s − 0.733·67-s − 0.118·71-s − 0.936·73-s − 1.35·79-s − 0.658·83-s − 0.433·85-s − 0.953·89-s + 0.512·95-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 \( 1 + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.70715175882258329308009960961, −6.99835950622058167840985689016, −6.08438344147867886231929340267, −5.64407299768507025484510862681, −4.86154103115382654376507449448, −4.02492507772792149421985969698, −3.08606864493245058459527762378, −2.34230964433500185499999339359, −1.37354983862931044783723956819, 0, 1.37354983862931044783723956819, 2.34230964433500185499999339359, 3.08606864493245058459527762378, 4.02492507772792149421985969698, 4.86154103115382654376507449448, 5.64407299768507025484510862681, 6.08438344147867886231929340267, 6.99835950622058167840985689016, 7.70715175882258329308009960961

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