Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s − 7-s + 9-s − 11-s − 6·13-s − 2·15-s + 2·17-s + 4·19-s + 21-s − 25-s − 27-s + 2·29-s − 8·31-s + 33-s − 2·35-s − 6·37-s + 6·39-s + 10·41-s − 4·43-s + 2·45-s + 8·47-s + 49-s − 2·51-s − 6·53-s − 2·55-s − 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 0.516·15-s + 0.485·17-s + 0.917·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.174·33-s − 0.338·35-s − 0.986·37-s + 0.960·39-s + 1.56·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 0.269·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(14784\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{14784} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 14784,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.397841719$
$L(\frac12)$  $\approx$  $1.397841719$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−16.17735134908903, −15.67011652067026, −14.84242566223750, −14.41421114810634, −13.85880300210027, −13.23287445028070, −12.63328151681418, −12.22957883422910, −11.65759749759883, −10.92304473248759, −10.21271726793200, −9.910454165306144, −9.379865054237293, −8.788812838692639, −7.602014615937485, −7.436793311717465, −6.677858412579330, −5.881056445612882, −5.424878863008108, −4.964954101900904, −4.066971997800218, −3.142205017540144, −2.407558880642260, −1.659975423315728, −0.5234685549957402, 0.5234685549957402, 1.659975423315728, 2.407558880642260, 3.142205017540144, 4.066971997800218, 4.964954101900904, 5.424878863008108, 5.881056445612882, 6.677858412579330, 7.436793311717465, 7.602014615937485, 8.788812838692639, 9.379865054237293, 9.910454165306144, 10.21271726793200, 10.92304473248759, 11.65759749759883, 12.22957883422910, 12.63328151681418, 13.23287445028070, 13.85880300210027, 14.41421114810634, 14.84242566223750, 15.67011652067026, 16.17735134908903

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