Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
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  − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 11-s − 2·13-s − 14-s + 16-s − 2·17-s − 4·19-s − 20-s − 22-s − 8·23-s + 25-s + 2·26-s + 28-s − 6·29-s + 8·31-s − 32-s + 2·34-s − 35-s − 2·37-s + 4·38-s + 40-s − 2·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.223·20-s − 0.213·22-s − 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 0.169·35-s − 0.328·37-s + 0.648·38-s + 0.158·40-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6930\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6930} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6930,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9075788965$
$L(\frac12)$  $\approx$  $0.9075788965$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−17.38673326453280, −16.53172935864057, −16.26266012593935, −15.37090668654501, −15.02994525974935, −14.40573445385016, −13.68674362198213, −12.97494741592471, −12.18945298491815, −11.74634648205164, −11.24347518227283, −10.36421910384271, −10.06310686780762, −9.158228125920084, −8.599406116944460, −7.973713173933111, −7.459358733286653, −6.621249185341335, −6.086648655109486, −5.100462879833416, −4.285690188341616, −3.629641232038573, −2.436908659351295, −1.836149785882708, −0.5253145380382009, 0.5253145380382009, 1.836149785882708, 2.436908659351295, 3.629641232038573, 4.285690188341616, 5.100462879833416, 6.086648655109486, 6.621249185341335, 7.459358733286653, 7.973713173933111, 8.599406116944460, 9.158228125920084, 10.06310686780762, 10.36421910384271, 11.24347518227283, 11.74634648205164, 12.18945298491815, 12.97494741592471, 13.68674362198213, 14.40573445385016, 15.02994525974935, 15.37090668654501, 16.26266012593935, 16.53172935864057, 17.38673326453280

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