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 − 8-s + 10-s + 11-s + 6·13-s + 16-s − 17-s + 4·19-s − 20-s − 22-s + 25-s − 6·26-s − 6·29-s − 32-s + 34-s − 2·37-s − 4·38-s + 40-s + 6·41-s − 4·43-s + 44-s + 8·47-s − 7·49-s − 50-s + 6·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 1.66·13-s + 1/4·16-s − 0.242·17-s + 0.917·19-s − 0.223·20-s − 0.213·22-s + 1/5·25-s − 1.17·26-s − 1.11·29-s − 0.176·32-s + 0.171·34-s − 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + 0.150·44-s + 1.16·47-s − 49-s − 0.141·50-s + 0.832·52-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 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
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 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−16.20916630671423, −15.59291439225709, −15.43839484156812, −14.50501619749561, −14.04455426680990, −13.39226085972240, −12.79877452297956, −12.20592562971697, −11.40459122346186, −11.22050518799189, −10.65802749273998, −9.919653178054412, −9.164493268754505, −8.956983650365216, −8.119014706168677, −7.712762118795212, −7.050896144251517, −6.299470393549461, −5.881455916175576, −5.012716301554793, −4.118597161963443, −3.505139891636901, −2.882878046777655, −1.701675196050300, −1.149203447131931, 0, 1.149203447131931, 1.701675196050300, 2.882878046777655, 3.505139891636901, 4.118597161963443, 5.012716301554793, 5.881455916175576, 6.299470393549461, 7.050896144251517, 7.712762118795212, 8.119014706168677, 8.956983650365216, 9.164493268754505, 9.919653178054412, 10.65802749273998, 11.22050518799189, 11.40459122346186, 12.20592562971697, 12.79877452297956, 13.39226085972240, 14.04455426680990, 14.50501619749561, 15.43839484156812, 15.59291439225709, 16.20916630671423

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