Properties

Degree $2$
Conductor $6534$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 3·5-s + 7-s + 8-s + 3·10-s + 4·13-s + 14-s + 16-s − 2·19-s + 3·20-s − 6·23-s + 4·25-s + 4·26-s + 28-s − 6·29-s + 5·31-s + 32-s + 3·35-s + 2·37-s − 2·38-s + 3·40-s + 6·41-s + 10·43-s − 6·46-s + 6·47-s − 6·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s + 0.353·8-s + 0.948·10-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.458·19-s + 0.670·20-s − 1.25·23-s + 4/5·25-s + 0.784·26-s + 0.188·28-s − 1.11·29-s + 0.898·31-s + 0.176·32-s + 0.507·35-s + 0.328·37-s − 0.324·38-s + 0.474·40-s + 0.937·41-s + 1.52·43-s − 0.884·46-s + 0.875·47-s − 6/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6534\)    =    \(2 \cdot 3^{3} \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{6534} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6534,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.843552356\)
\(L(\frac12)\) \(\approx\) \(4.843552356\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−8.015255053006877536003075086198, −7.12468139474762615848955209063, −6.29574961437633563574123917046, −5.84315760589664083898483470884, −5.39074241880952165115421872247, −4.30456169908575776721292993281, −3.80538107085536458846182311961, −2.58138575780524151090389938489, −2.03246862806699813641002498927, −1.09285120951784850918046733268, 1.09285120951784850918046733268, 2.03246862806699813641002498927, 2.58138575780524151090389938489, 3.80538107085536458846182311961, 4.30456169908575776721292993281, 5.39074241880952165115421872247, 5.84315760589664083898483470884, 6.29574961437633563574123917046, 7.12468139474762615848955209063, 8.015255053006877536003075086198

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