Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 $
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 + 8-s + 11-s + 4·13-s + 16-s − 6·17-s + 4·19-s + 22-s − 6·23-s − 5·25-s + 4·26-s − 6·29-s − 8·31-s + 32-s − 6·34-s − 10·37-s + 4·38-s + 6·41-s + 8·43-s + 44-s − 6·46-s − 6·47-s − 5·50-s + 4·52-s − 6·58-s − 8·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.301·11-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.213·22-s − 1.25·23-s − 25-s + 0.784·26-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s − 1.64·37-s + 0.648·38-s + 0.937·41-s + 1.21·43-s + 0.150·44-s − 0.884·46-s − 0.875·47-s − 0.707·50-s + 0.554·52-s − 0.787·58-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9702\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9702} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 9702,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
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

−16.82913380104622, −16.14685438996098, −15.80865164464699, −15.35137907628051, −14.56306405639086, −14.00356390255431, −13.54571578646819, −13.07436309856914, −12.34085557971961, −11.79714856914786, −11.02687961162561, −10.92086943522125, −9.907066535635558, −9.222166695435719, −8.714298869752204, −7.819406919360613, −7.290879869758473, −6.525048294383817, −5.859554622950202, −5.446879105275632, −4.363769643931715, −3.897614579929360, −3.252611580601609, −2.120207219284307, −1.541243699177946, 0, 1.541243699177946, 2.120207219284307, 3.252611580601609, 3.897614579929360, 4.363769643931715, 5.446879105275632, 5.859554622950202, 6.525048294383817, 7.290879869758473, 7.819406919360613, 8.714298869752204, 9.222166695435719, 9.907066535635558, 10.92086943522125, 11.02687961162561, 11.79714856914786, 12.34085557971961, 13.07436309856914, 13.54571578646819, 14.00356390255431, 14.56306405639086, 15.35137907628051, 15.80865164464699, 16.14685438996098, 16.82913380104622

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