Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 11 \cdot 43 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 2·7-s + 8-s − 2·9-s − 10-s − 11-s + 12-s − 4·13-s + 2·14-s − 15-s + 16-s − 2·18-s + 2·19-s − 20-s + 2·21-s − 22-s − 6·23-s + 24-s + 25-s − 4·26-s − 5·27-s + 2·28-s − 9·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 1.10·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.471·18-s + 0.458·19-s − 0.223·20-s + 0.436·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.784·26-s − 0.962·27-s + 0.377·28-s − 1.67·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4730\)    =    \(2 \cdot 5 \cdot 11 \cdot 43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4730} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4730,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 + T \)
43 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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

−7.79192681673931850704308557203, −7.45420020677640366074606581187, −6.45710724352738035215939020242, −5.40234671614346677776265259812, −5.12485665700417171281648483978, −4.03982079845454910269028067547, −3.45088142905244326918274351752, −2.47091054541652952161291115006, −1.81363527596428679583875624632, 0, 1.81363527596428679583875624632, 2.47091054541652952161291115006, 3.45088142905244326918274351752, 4.03982079845454910269028067547, 5.12485665700417171281648483978, 5.40234671614346677776265259812, 6.45710724352738035215939020242, 7.45420020677640366074606581187, 7.79192681673931850704308557203

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