Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s + 11-s − 12-s − 4·13-s + 2·14-s + 16-s − 18-s − 4·19-s + 2·21-s − 22-s − 6·23-s + 24-s − 5·25-s + 4·26-s − 27-s − 2·28-s − 6·29-s − 8·31-s − 32-s − 33-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.436·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 25-s + 0.784·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.174·33-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19074\)    =    \(2 \cdot 3 \cdot 11 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19074} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 19074,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−16.36978681565520, −15.98725737611398, −15.19072270284399, −14.82255814833627, −14.21362234249453, −13.39657926027670, −12.69683589703553, −12.54116016214777, −11.66390325128082, −11.37426971519441, −10.64984028471100, −10.00198791121838, −9.656474939094952, −9.184332047364308, −8.382393395669576, −7.621003635533524, −7.296308151012698, −6.519459824769081, −5.955263508995293, −5.544474635033775, −4.448406186035721, −3.969426125228993, −3.041087480946066, −2.176375877602884, −1.543641643471946, 0, 0, 1.543641643471946, 2.176375877602884, 3.041087480946066, 3.969426125228993, 4.448406186035721, 5.544474635033775, 5.955263508995293, 6.519459824769081, 7.296308151012698, 7.621003635533524, 8.382393395669576, 9.184332047364308, 9.656474939094952, 10.00198791121838, 10.64984028471100, 11.37426971519441, 11.66390325128082, 12.54116016214777, 12.69683589703553, 13.39657926027670, 14.21362234249453, 14.82255814833627, 15.19072270284399, 15.98725737611398, 16.36978681565520

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