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 + 6·13-s + 15-s + 2·17-s + 8·19-s − 21-s + 8·23-s + 25-s − 27-s + 2·29-s + 4·31-s − 35-s + 2·37-s − 6·39-s − 6·41-s − 4·43-s − 45-s + 8·47-s + 49-s − 2·51-s − 10·53-s − 8·57-s − 4·59-s + 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.258·15-s + 0.485·17-s + 1.83·19-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 + 0.328·37-s − 0.960·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s − 1.05·57-s − 0.520·59-s + 0.256·61-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  =  $(2,\ 6720,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.073043849$
$L(\frac12)$  $\approx$  $2.073043849$
$L(\frac{3}{2})$   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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−17.22166125861003, −16.59984419420073, −16.12769373775325, −15.46990671374131, −15.17272251524346, −14.12446741675780, −13.74324532612387, −13.07396329886774, −12.39437765550026, −11.52390290248182, −11.49820660940229, −10.68660202310020, −10.10568478989214, −9.216738438201368, −8.662057175587300, −7.890343804515540, −7.310644862230137, −6.560817911347647, −5.838097154208113, −5.141665055489783, −4.518663056842299, −3.498201920693507, −3.013413898764592, −1.420999045736720, −0.8858141808045819, 0.8858141808045819, 1.420999045736720, 3.013413898764592, 3.498201920693507, 4.518663056842299, 5.141665055489783, 5.838097154208113, 6.560817911347647, 7.310644862230137, 7.890343804515540, 8.662057175587300, 9.216738438201368, 10.10568478989214, 10.68660202310020, 11.49820660940229, 11.52390290248182, 12.39437765550026, 13.07396329886774, 13.74324532612387, 14.12446741675780, 15.17272251524346, 15.46990671374131, 16.12769373775325, 16.59984419420073, 17.22166125861003

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