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  − 7-s − 6·11-s + 2·13-s + 4·19-s − 6·23-s − 5·25-s − 6·29-s − 8·31-s + 2·37-s − 12·41-s + 4·43-s + 12·47-s + 49-s + 6·53-s − 10·61-s − 8·67-s + 6·71-s − 10·73-s + 6·77-s + 4·79-s − 12·83-s − 12·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.80·11-s + 0.554·13-s + 0.917·19-s − 1.25·23-s − 25-s − 1.11·29-s − 1.43·31-s + 0.328·37-s − 1.87·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 1.28·61-s − 0.977·67-s + 0.712·71-s − 1.17·73-s + 0.683·77-s + 0.450·79-s − 1.31·83-s − 1.27·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−18.96794579396369, −18.24541640623415, −18.12562184690893, −16.92696710923473, −16.22903383884115, −15.64161597199658, −15.12696153453565, −13.90135335283811, −13.49581850505824, −12.75070592435053, −11.95274200206902, −11.05654753058707, −10.33314640525589, −9.666722228038389, −8.701525582792892, −7.760511522009152, −7.260310817113322, −5.822776200681514, −5.480366432706380, −4.113999371142057, −3.128753725700454, −1.962577435188673, 0, 1.962577435188673, 3.128753725700454, 4.113999371142057, 5.480366432706380, 5.822776200681514, 7.260310817113322, 7.760511522009152, 8.701525582792892, 9.666722228038389, 10.33314640525589, 11.05654753058707, 11.95274200206902, 12.75070592435053, 13.49581850505824, 13.90135335283811, 15.12696153453565, 15.64161597199658, 16.22903383884115, 16.92696710923473, 18.12562184690893, 18.24541640623415, 18.96794579396369

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