Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5 \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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 + 8·23-s + 25-s − 27-s − 6·29-s + 8·31-s − 4·33-s + 35-s + 2·37-s − 2·39-s + 2·41-s − 12·43-s − 45-s + 8·47-s + 49-s − 2·51-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 + 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 − 1.11·29-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 − 1.82·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-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  =  0
Selberg data  =  $(2,\ 6720,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.545373716$
$L(\frac12)$  $\approx$  $1.545373716$
$L(\frac{3}{2})$   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 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−17.13130598162396, −16.80916022375753, −16.13198276510460, −15.55500414867294, −14.77583804863119, −14.60839437877681, −13.35571930167247, −13.24223628127073, −12.28139701257463, −11.90655462967914, −11.20054528667871, −10.77943836745859, −9.961661818899658, −9.290434134060844, −8.696290074873709, −7.990738894141251, −7.045701912035120, −6.629047025952524, −5.988051524177010, −5.142671464753175, −4.321085365634661, −3.711138445434791, −2.887145202894325, −1.575917879652175, −0.6992687704392115, 0.6992687704392115, 1.575917879652175, 2.887145202894325, 3.711138445434791, 4.321085365634661, 5.142671464753175, 5.988051524177010, 6.629047025952524, 7.045701912035120, 7.990738894141251, 8.696290074873709, 9.290434134060844, 9.961661818899658, 10.77943836745859, 11.20054528667871, 11.90655462967914, 12.28139701257463, 13.24223628127073, 13.35571930167247, 14.60839437877681, 14.77583804863119, 15.55500414867294, 16.13198276510460, 16.80916022375753, 17.13130598162396

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