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 − 2·13-s + 15-s − 2·17-s − 4·19-s + 21-s + 8·23-s + 25-s − 27-s + 2·29-s + 4·31-s + 35-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s − 45-s + 49-s + 2·51-s − 6·53-s + 4·57-s − 6·61-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 − 0.485·17-s − 0.917·19-s + 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.169·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.529·57-s − 0.768·61-s − 0.125·63-s + 0.248·65-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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
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 \( 1 + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−17.43485717039055, −16.87272725951938, −16.58173971649813, −15.67894861432256, −15.28076232422385, −14.79739429546430, −13.93975281723685, −13.24745206375159, −12.68003070229811, −12.23737970773408, −11.48299542862004, −10.92663727944609, −10.44480111154991, −9.593603813745660, −9.067070003283773, −8.247996468551095, −7.598092135321996, −6.665212987814385, −6.526992870073626, −5.426915248979014, −4.749383675997965, −4.162685181844502, −3.152871252162442, −2.370570903960267, −1.082832173353418, 0, 1.082832173353418, 2.370570903960267, 3.152871252162442, 4.162685181844502, 4.749383675997965, 5.426915248979014, 6.526992870073626, 6.665212987814385, 7.598092135321996, 8.247996468551095, 9.067070003283773, 9.593603813745660, 10.44480111154991, 10.92663727944609, 11.48299542862004, 12.23737970773408, 12.68003070229811, 13.24745206375159, 13.93975281723685, 14.79739429546430, 15.28076232422385, 15.67894861432256, 16.58173971649813, 16.87272725951938, 17.43485717039055

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