Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 11 \cdot 43 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.743·3-s + 4-s − 5-s − 0.743·6-s + 2.37·7-s + 8-s − 2.44·9-s − 10-s + 11-s − 0.743·12-s − 6.39·13-s + 2.37·14-s + 0.743·15-s + 16-s + 4.07·17-s − 2.44·18-s + 0.435·19-s − 20-s − 1.76·21-s + 22-s + 6.26·23-s − 0.743·24-s + 25-s − 6.39·26-s + 4.05·27-s + 2.37·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.429·3-s + 0.5·4-s − 0.447·5-s − 0.303·6-s + 0.895·7-s + 0.353·8-s − 0.815·9-s − 0.316·10-s + 0.301·11-s − 0.214·12-s − 1.77·13-s + 0.633·14-s + 0.192·15-s + 0.250·16-s + 0.989·17-s − 0.576·18-s + 0.0998·19-s − 0.223·20-s − 0.384·21-s + 0.213·22-s + 1.30·23-s − 0.151·24-s + 0.200·25-s − 1.25·26-s + 0.779·27-s + 0.447·28-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  =  0
Selberg data  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.343237507$
$L(\frac12)$  $\approx$  $2.343237507$
$L(\frac{3}{2})$   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 \( 1 + 0.743T + 3T^{2} \)
7 \( 1 - 2.37T + 7T^{2} \)
13 \( 1 + 6.39T + 13T^{2} \)
17 \( 1 - 4.07T + 17T^{2} \)
19 \( 1 - 0.435T + 19T^{2} \)
23 \( 1 - 6.26T + 23T^{2} \)
29 \( 1 + 1.79T + 29T^{2} \)
31 \( 1 + 8.03T + 31T^{2} \)
37 \( 1 - 2.46T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
47 \( 1 - 13.4T + 47T^{2} \)
53 \( 1 + 3.17T + 53T^{2} \)
59 \( 1 - 3.31T + 59T^{2} \)
61 \( 1 + 7.47T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 - 3.81T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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

−8.004249576855716221407204789488, −7.50842649949025817606100055984, −6.91075615269052869804294096159, −5.86389061911553257107252130751, −5.18085252288294772306574707282, −4.85243856989569142909130174683, −3.84061677560699225519455031331, −2.95405236984059302182977754360, −2.10939711003906668129235593738, −0.77082066375741336096172630543, 0.77082066375741336096172630543, 2.10939711003906668129235593738, 2.95405236984059302182977754360, 3.84061677560699225519455031331, 4.85243856989569142909130174683, 5.18085252288294772306574707282, 5.86389061911553257107252130751, 6.91075615269052869804294096159, 7.50842649949025817606100055984, 8.004249576855716221407204789488

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