Properties

Degree 2
Conductor $ 2^{6} \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 − 4·11-s + 2·13-s + 15-s + 2·17-s − 4·19-s + 21-s + 25-s − 27-s + 2·29-s + 8·31-s + 4·33-s + 35-s + 2·37-s − 2·39-s + 2·41-s − 4·43-s − 45-s + 49-s − 2·51-s + 10·53-s + 4·55-s + 4·57-s + 12·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-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 + 0.696·33-s + 0.169·35-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s + 1.37·53-s + 0.539·55-s + 0.529·57-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6720\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6720} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6720,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−17.61705075113826, −16.81171476604556, −16.40422299104516, −15.77377352251104, −15.37005718764195, −14.75060395867508, −13.88096090896536, −13.17961010813309, −12.88955105396940, −12.04925993647820, −11.63964298925666, −10.83129077583890, −10.31000966800398, −9.912750581370814, −8.811324944381096, −8.309892688630534, −7.614651131493761, −6.909400382202342, −6.164585575616498, −5.596137224685135, −4.742224494235028, −4.116645675232456, −3.158283423335220, −2.388734757870861, −1.068854188289235, 0, 1.068854188289235, 2.388734757870861, 3.158283423335220, 4.116645675232456, 4.742224494235028, 5.596137224685135, 6.164585575616498, 6.909400382202342, 7.614651131493761, 8.309892688630534, 8.811324944381096, 9.912750581370814, 10.31000966800398, 10.83129077583890, 11.63964298925666, 12.04925993647820, 12.88955105396940, 13.17961010813309, 13.88096090896536, 14.75060395867508, 15.37005718764195, 15.77377352251104, 16.40422299104516, 16.81171476604556, 17.61705075113826

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