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 + 6·13-s + 15-s + 6·17-s − 4·19-s + 21-s + 4·23-s + 25-s − 27-s + 2·29-s − 8·31-s + 4·33-s + 35-s − 6·37-s − 6·39-s + 6·41-s + 8·43-s − 45-s + 49-s − 6·51-s − 6·53-s + 4·55-s + 4·57-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 + 0.834·23-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 − 0.986·37-s − 0.960·39-s + 0.937·41-s + 1.21·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.539·55-s + 0.529·57-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  =  $(2,\ 6720,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.209130472$
$L(\frac12)$  $\approx$  $1.209130472$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−17.19483118458092, −16.55470594546896, −15.99704330369008, −15.75982253925407, −15.00147062924493, −14.31657373650763, −13.57992702149478, −12.80019317974300, −12.70151715458698, −11.85478933156007, −11.02448929954641, −10.70926656369216, −10.22890912972491, −9.189837492959954, −8.695877598667159, −7.791095745553468, −7.433866239809372, −6.442784605936017, −5.858586781988541, −5.283119787181863, −4.386542786705784, −3.557387931391657, −2.945843364770567, −1.638011147204542, −0.5961194499063336, 0.5961194499063336, 1.638011147204542, 2.945843364770567, 3.557387931391657, 4.386542786705784, 5.283119787181863, 5.858586781988541, 6.442784605936017, 7.433866239809372, 7.791095745553468, 8.695877598667159, 9.189837492959954, 10.22890912972491, 10.70926656369216, 11.02448929954641, 11.85478933156007, 12.70151715458698, 12.80019317974300, 13.57992702149478, 14.31657373650763, 15.00147062924493, 15.75982253925407, 15.99704330369008, 16.55470594546896, 17.19483118458092

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