Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s − 4·11-s + 2·13-s + 6·17-s − 8·19-s − 25-s − 6·29-s − 8·31-s − 2·35-s − 2·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 6·53-s + 8·55-s − 6·61-s − 4·65-s + 4·67-s − 8·71-s + 10·73-s − 4·77-s − 16·79-s + 8·83-s − 12·85-s + 6·89-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s − 0.768·61-s − 0.496·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.455·77-s − 1.80·79-s + 0.878·83-s − 1.30·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.81091954122851, −18.84302431198750, −18.76537182946652, −17.80542961159119, −16.92676191659814, −16.30219183151636, −15.59822957511555, −14.91621165711342, −14.36962942814042, −13.22122852644190, −12.72249388729552, −11.92211761661144, −10.98784923157337, −10.63145032630949, −9.559817840156688, −8.471445676889714, −7.930145034161771, −7.275542518020635, −6.021997044143567, −5.194033396068595, −4.123066989008585, −3.275995389802918, −1.865371178942683, 0, 1.865371178942683, 3.275995389802918, 4.123066989008585, 5.194033396068595, 6.021997044143567, 7.275542518020635, 7.930145034161771, 8.471445676889714, 9.559817840156688, 10.63145032630949, 10.98784923157337, 11.92211761661144, 12.72249388729552, 13.22122852644190, 14.36962942814042, 14.91621165711342, 15.59822957511555, 16.30219183151636, 16.92676191659814, 17.80542961159119, 18.76537182946652, 18.84302431198750, 19.81091954122851

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