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 − 13-s − 14-s + 16-s + 17-s − 5·19-s − 20-s − 22-s − 9·23-s + 25-s + 26-s + 28-s + 4·29-s − 7·31-s − 32-s − 34-s − 35-s + 7·37-s + 5·38-s + 40-s + 10·41-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 − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.14·19-s − 0.223·20-s − 0.213·22-s − 1.87·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s + 0.742·29-s − 1.25·31-s − 0.176·32-s − 0.171·34-s − 0.169·35-s + 1.15·37-s + 0.811·38-s + 0.158·40-s + 1.56·41-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  =  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 \( 1 + T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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

−16.38139264042273, −15.75344772749971, −14.98541381103441, −14.70159844523315, −14.13462702719716, −13.41148685560561, −12.61423509404127, −12.22584848361661, −11.68155583912759, −10.99068019171104, −10.67492365549038, −9.833299232663071, −9.491296491716443, −8.656297870209069, −8.148437632376131, −7.775628247359625, −7.025918737161001, −6.384764369084290, −5.803314749192487, −4.972294039679647, −4.102443664525354, −3.725844194003188, −2.521720744606798, −2.035082510537466, −0.9918611256015191, 0, 0.9918611256015191, 2.035082510537466, 2.521720744606798, 3.725844194003188, 4.102443664525354, 4.972294039679647, 5.803314749192487, 6.384764369084290, 7.025918737161001, 7.775628247359625, 8.148437632376131, 8.656297870209069, 9.491296491716443, 9.833299232663071, 10.67492365549038, 10.99068019171104, 11.68155583912759, 12.22584848361661, 12.61423509404127, 13.41148685560561, 14.13462702719716, 14.70159844523315, 14.98541381103441, 15.75344772749971, 16.38139264042273

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