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 1

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 − 4·13-s − 14-s + 16-s − 17-s − 2·19-s − 20-s + 22-s + 3·23-s + 25-s + 4·26-s + 28-s − 7·29-s − 5·31-s − 32-s + 34-s − 35-s + 2·38-s + 40-s + 10·41-s − 3·43-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 − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s − 0.223·20-s + 0.213·22-s + 0.625·23-s + 1/5·25-s + 0.784·26-s + 0.188·28-s − 1.29·29-s − 0.898·31-s − 0.176·32-s + 0.171·34-s − 0.169·35-s + 0.324·38-s + 0.158·40-s + 1.56·41-s − 0.457·43-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  =  1
Selberg data  =  $(2,\ 16830,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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

−16.16628620111521, −15.72205876451788, −15.07938116427193, −14.58222139200148, −14.29288406826750, −13.15923999729795, −12.83018542981852, −12.26447405401970, −11.46519026091841, −11.18320004057239, −10.58962743231496, −9.933004020153550, −9.297493984664914, −8.880559069029173, −8.046124731155651, −7.693242812346355, −7.061395099072256, −6.565046556434025, −5.482431504619217, −5.165768100539495, −4.189527320236771, −3.598013205449691, −2.500788258005834, −2.121896807196039, −0.9364391349799294, 0, 0.9364391349799294, 2.121896807196039, 2.500788258005834, 3.598013205449691, 4.189527320236771, 5.165768100539495, 5.482431504619217, 6.565046556434025, 7.061395099072256, 7.693242812346355, 8.046124731155651, 8.880559069029173, 9.297493984664914, 9.933004020153550, 10.58962743231496, 11.18320004057239, 11.46519026091841, 12.26447405401970, 12.83018542981852, 13.15923999729795, 14.29288406826750, 14.58222139200148, 15.07938116427193, 15.72205876451788, 16.16628620111521

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