Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13 $
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 − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s − 13-s + 14-s + 16-s − 6·17-s − 18-s − 4·19-s + 21-s + 24-s + 26-s − 27-s − 28-s + 6·29-s − 4·31-s − 32-s + 6·34-s + 36-s + 10·37-s + 4·38-s + 39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.218·21-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.64·37-s + 0.648·38-s + 0.160·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{13650} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 13650,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
13 \( 1 + T \)
good11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−16.53577972574953, −16.06042173879892, −15.46240217490684, −14.98258435215051, −14.37378789391304, −13.47768167753301, −13.00110603842114, −12.50639978362969, −11.78537174570469, −11.27142017810084, −10.72856954948314, −10.24505661118163, −9.548189673522124, −9.033544564490981, −8.408822302098008, −7.739885161079115, −7.001319631707097, −6.438020498655731, −6.083471804991526, −5.079527152899175, −4.453031973045118, −3.700649172011494, −2.602314594017692, −2.069974889439539, −0.8973555198840396, 0, 0.8973555198840396, 2.069974889439539, 2.602314594017692, 3.700649172011494, 4.453031973045118, 5.079527152899175, 6.083471804991526, 6.438020498655731, 7.001319631707097, 7.739885161079115, 8.408822302098008, 9.033544564490981, 9.548189673522124, 10.24505661118163, 10.72856954948314, 11.27142017810084, 11.78537174570469, 12.50639978362969, 13.00110603842114, 13.47768167753301, 14.37378789391304, 14.98258435215051, 15.46240217490684, 16.06042173879892, 16.53577972574953

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