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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s + 9-s − 2·11-s − 4·13-s + 15-s + 2·17-s + 2·19-s + 21-s − 4·23-s + 25-s − 27-s − 6·29-s + 2·31-s + 2·33-s + 35-s − 10·37-s + 4·39-s − 10·41-s + 12·43-s − 45-s + 8·47-s + 49-s − 2·51-s + 2·55-s − 2·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.258·15-s + 0.485·17-s + 0.458·19-s + 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.348·33-s + 0.169·35-s − 1.64·37-s + 0.640·39-s − 1.56·41-s + 1.82·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.269·55-s − 0.264·57-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  =  $(2,\ 6720,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6923238545$
$L(\frac12)$  $\approx$  $0.6923238545$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.18299130622772, −16.70606966355303, −16.09611960129180, −15.46165539946063, −15.16993636393987, −14.15175555887711, −13.80733499188321, −12.88638458179842, −12.30689149909820, −12.03381111737500, −11.26397950794031, −10.51399219233998, −10.08224444683866, −9.414170181756123, −8.652798549105795, −7.686447322260534, −7.411339950062814, −6.643506778108549, −5.705221438407930, −5.272647720683379, −4.436217061659737, −3.633573307155716, −2.790013392686649, −1.789655425572838, −0.4258873900958938, 0.4258873900958938, 1.789655425572838, 2.790013392686649, 3.633573307155716, 4.436217061659737, 5.272647720683379, 5.705221438407930, 6.643506778108549, 7.411339950062814, 7.686447322260534, 8.652798549105795, 9.414170181756123, 10.08224444683866, 10.51399219233998, 11.26397950794031, 12.03381111737500, 12.30689149909820, 12.88638458179842, 13.80733499188321, 14.15175555887711, 15.16993636393987, 15.46165539946063, 16.09611960129180, 16.70606966355303, 17.18299130622772

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