Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 7-s + 9-s + 15-s − 6·17-s + 8·19-s − 21-s + 25-s − 27-s + 6·29-s − 4·31-s − 35-s + 10·37-s + 6·41-s + 4·43-s − 45-s + 49-s + 6·51-s − 6·53-s − 8·57-s − 12·59-s − 10·61-s + 63-s − 4·67-s + 12·71-s + 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.258·15-s − 1.45·17-s + 1.83·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.169·35-s + 1.64·37-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 1.05·57-s − 1.56·59-s − 1.28·61-s + 0.125·63-s − 0.488·67-s + 1.42·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(283920\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{283920} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 283920,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.96909614337919, −12.32027403926367, −12.06871444475031, −11.57033939115958, −11.01765418613959, −10.93138420595349, −10.39259475413450, −9.528363814242661, −9.400475517756277, −8.942232234001357, −8.171414128079034, −7.712265228931457, −7.533475986485649, −6.783650425444773, −6.407584847669064, −5.891769316586545, −5.331681875613937, −4.718358923928472, −4.520046203452877, −3.891681123633268, −3.224917631636464, −2.687729664171328, −2.068771681554196, −1.251754158993490, −0.8077019329449754, 0, 0.8077019329449754, 1.251754158993490, 2.068771681554196, 2.687729664171328, 3.224917631636464, 3.891681123633268, 4.520046203452877, 4.718358923928472, 5.331681875613937, 5.891769316586545, 6.407584847669064, 6.783650425444773, 7.533475986485649, 7.712265228931457, 8.171414128079034, 8.942232234001357, 9.400475517756277, 9.528363814242661, 10.39259475413450, 10.93138420595349, 11.01765418613959, 11.57033939115958, 12.06871444475031, 12.32027403926367, 12.96909614337919

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