Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \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  − 3-s + 5-s − 7-s + 9-s + 2·13-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 − 2·39-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 + 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-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 − 0.320·39-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 + 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1680} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1680,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, 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 \( 1 + T \)
good11 \( 1 + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
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

−19.27773473447021, −18.84514590496468, −17.90913480192045, −17.42436039105841, −16.99354154425237, −15.98383814988374, −15.63444315798779, −14.83720742408199, −13.89120546503642, −13.29862717757556, −12.70544141126070, −11.96694186216097, −11.07791237625902, −10.52187360021345, −9.916721087133813, −8.746285736623813, −8.508034972112431, −7.025200281718321, −6.508499003058611, −5.882529793468564, −4.766057608719710, −4.083880452418900, −2.752930927523867, −1.655900987002804, 0, 1.655900987002804, 2.752930927523867, 4.083880452418900, 4.766057608719710, 5.882529793468564, 6.508499003058611, 7.025200281718321, 8.508034972112431, 8.746285736623813, 9.916721087133813, 10.52187360021345, 11.07791237625902, 11.96694186216097, 12.70544141126070, 13.29862717757556, 13.89120546503642, 14.83720742408199, 15.63444315798779, 15.98383814988374, 16.99354154425237, 17.42436039105841, 17.90913480192045, 18.84514590496468, 19.27773473447021

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