Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17 $
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 + 3·7-s − 8-s + 10-s + 11-s + 4·13-s − 3·14-s + 16-s − 17-s + 6·19-s − 20-s − 22-s − 23-s + 25-s − 4·26-s + 3·28-s − 9·29-s + 7·31-s − 32-s + 34-s − 3·35-s − 4·37-s − 6·38-s + 40-s − 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 1.10·13-s − 0.801·14-s + 1/4·16-s − 0.242·17-s + 1.37·19-s − 0.223·20-s − 0.213·22-s − 0.208·23-s + 1/5·25-s − 0.784·26-s + 0.566·28-s − 1.67·29-s + 1.25·31-s − 0.176·32-s + 0.171·34-s − 0.507·35-s − 0.657·37-s − 0.973·38-s + 0.158·40-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16830\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{16830} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 16830,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.819136338$
$L(\frac12)$  $\approx$  $1.819136338$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;11,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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

−15.88215449924090, −15.47867849809264, −14.86892409908958, −14.36371505000601, −13.68976135664766, −13.23347001082771, −12.39518091676603, −11.59084119141768, −11.46495360193275, −11.06650051811759, −10.15537011284201, −9.737526077444154, −8.960480099037326, −8.287986775000248, −8.148693833333493, −7.354997138484072, −6.768110745077172, −6.092209554826420, −5.208785572306025, −4.802659638228375, −3.635008347687522, −3.414506303691761, −2.100661908377849, −1.512230748377804, −0.6840349300533937, 0.6840349300533937, 1.512230748377804, 2.100661908377849, 3.414506303691761, 3.635008347687522, 4.802659638228375, 5.208785572306025, 6.092209554826420, 6.768110745077172, 7.354997138484072, 8.148693833333493, 8.287986775000248, 8.960480099037326, 9.737526077444154, 10.15537011284201, 11.06650051811759, 11.46495360193275, 11.59084119141768, 12.39518091676603, 13.23347001082771, 13.68976135664766, 14.36371505000601, 14.86892409908958, 15.47867849809264, 15.88215449924090

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