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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.515·3-s + 4-s + 5-s + 0.515·6-s − 4.64·7-s − 8-s − 2.73·9-s − 10-s + 11-s − 0.515·12-s − 0.108·13-s + 4.64·14-s − 0.515·15-s + 16-s + 4.19·17-s + 2.73·18-s − 3.02·19-s + 20-s + 2.39·21-s − 22-s − 2.10·23-s + 0.515·24-s + 25-s + 0.108·26-s + 2.95·27-s − 4.64·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.297·3-s + 0.5·4-s + 0.447·5-s + 0.210·6-s − 1.75·7-s − 0.353·8-s − 0.911·9-s − 0.316·10-s + 0.301·11-s − 0.148·12-s − 0.0301·13-s + 1.24·14-s − 0.133·15-s + 0.250·16-s + 1.01·17-s + 0.644·18-s − 0.693·19-s + 0.223·20-s + 0.522·21-s − 0.213·22-s − 0.439·23-s + 0.105·24-s + 0.200·25-s + 0.0213·26-s + 0.568·27-s − 0.877·28-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  =  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 \( 1 + 0.515T + 3T^{2} \)
7 \( 1 + 4.64T + 7T^{2} \)
13 \( 1 + 0.108T + 13T^{2} \)
17 \( 1 - 4.19T + 17T^{2} \)
19 \( 1 + 3.02T + 19T^{2} \)
23 \( 1 + 2.10T + 23T^{2} \)
29 \( 1 - 8.06T + 29T^{2} \)
31 \( 1 - 3.88T + 31T^{2} \)
37 \( 1 - 6.82T + 37T^{2} \)
41 \( 1 - 2.16T + 41T^{2} \)
47 \( 1 + 9.78T + 47T^{2} \)
53 \( 1 - 1.84T + 53T^{2} \)
59 \( 1 - 1.63T + 59T^{2} \)
61 \( 1 - 8.07T + 61T^{2} \)
67 \( 1 + 3.43T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 - 7.68T + 73T^{2} \)
79 \( 1 - 6.50T + 79T^{2} \)
83 \( 1 + 6.38T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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

−8.146251078814364935251944943691, −7.07936206316730586929065877391, −6.27748883339790896859153140283, −6.18986998668251379091173882039, −5.24049189349218483033428422141, −4.01575428254930168172695728722, −3.01705212332249184109349189232, −2.57486162456305705307123485431, −1.08125724014400599212231258384, 0, 1.08125724014400599212231258384, 2.57486162456305705307123485431, 3.01705212332249184109349189232, 4.01575428254930168172695728722, 5.24049189349218483033428422141, 6.18986998668251379091173882039, 6.27748883339790896859153140283, 7.07936206316730586929065877391, 8.146251078814364935251944943691

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