Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5 \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s + 9-s − 4·11-s − 6·13-s + 15-s − 6·17-s − 4·19-s + 21-s − 8·23-s + 25-s − 27-s − 10·29-s + 4·31-s + 4·33-s + 35-s + 6·37-s + 6·39-s + 6·41-s − 4·43-s − 45-s − 12·47-s + 49-s + 6·51-s − 6·53-s + 4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.696·33-s + 0.169·35-s + 0.986·37-s + 0.960·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 0.539·55-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  =  2
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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−17.84195275838015, −17.24803982580957, −16.65863985829037, −16.03406637356921, −15.61910370646597, −14.90740407043061, −14.53316159948923, −13.39758545683372, −13.04031032198447, −12.51644350087593, −11.82795005789779, −11.23793701467463, −10.64164386406356, −9.955559051953748, −9.510374436822243, −8.573765101612345, −7.722736959970261, −7.470354409731500, −6.449084040093029, −6.014278610858485, −4.939940799702785, −4.598948481454358, −3.714950680317904, −2.550402766316839, −2.025632934217343, 0, 0, 2.025632934217343, 2.550402766316839, 3.714950680317904, 4.598948481454358, 4.939940799702785, 6.014278610858485, 6.449084040093029, 7.470354409731500, 7.722736959970261, 8.573765101612345, 9.510374436822243, 9.955559051953748, 10.64164386406356, 11.23793701467463, 11.82795005789779, 12.51644350087593, 13.04031032198447, 13.39758545683372, 14.53316159948923, 14.90740407043061, 15.61910370646597, 16.03406637356921, 16.65863985829037, 17.24803982580957, 17.84195275838015

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